A000984 - OEIS

%I M1645 N0643 #1056 Nov 04 2024 07:41:46

%S 1,2,6,20,70,252,924,3432,12870,48620,184756,705432,2704156,10400600,

%T 40116600,155117520,601080390,2333606220,9075135300,35345263800,

%U 137846528820,538257874440,2104098963720,8233430727600,32247603683100,126410606437752,495918532948104,1946939425648112

%N Central binomial coefficients: binomial(2*n,n) = (2*n)!/(n!)^2.

%C Devadoss refers to these numbers as type B Catalan numbers (cf. A000108).

%C Equal to the binomial coefficient sum Sum_{k=0..n} binomial(n,k)^2.

%C Number of possible interleavings of a program with n atomic instructions when executed by two processes. - Manuel Carro (mcarro(AT)fi.upm.es), Sep 22 2001

%C Convolving a(n) with itself yields A000302, the powers of 4. - _T. D. Noe_, Jun 11 2002

%C Number of ordered trees with 2n+1 edges, having root of odd degree and nonroot nodes of outdegree 0 or 2. - _Emeric Deutsch_, Aug 02 2002

%C Also number of directed, convex polyominoes having semiperimeter n+2.

%C Also number of diagonally symmetric, directed, convex polyominoes having semiperimeter 2n+2. - _Emeric Deutsch_, Aug 03 2002

%C The second inverse binomial transform of this sequence is this sequence with interpolated zeros. Its g.f. is (1 - 4*x^2)^(-1/2), with n-th term C(n,n/2)(1+(-1)^n)/2. - _Paul Barry_, Jul 01 2003

%C Number of possible values of a 2n-bit binary number for which half the bits are on and half are off. - Gavin Scott (gavin(AT)allegro.com), Aug 09 2003

%C Ordered partitions of n with zeros to n+1, e.g., for n=4 we consider the ordered partitions of 11110 (5), 11200 (30), 13000 (20), 40000 (5) and 22000 (10), total 70 and a(4)=70. See A001700 (esp. Mambetov Bektur's comment). - _Jon Perry_, Aug 10 2003

%C Number of nondecreasing sequences of n integers from 0 to n: a(n) = Sum_{i_1=0..n} Sum_{i_2=i_1..n}...Sum_{i_n=i_{n-1}..n}(1). - J. N. Bearden (jnb(AT)eller.arizona.edu), Sep 16 2003

%C Number of peaks at odd level in all Dyck paths of semilength n+1. Example: a(2)=6 because we have U*DU*DU*D, U*DUUDD, UUDDU*D, UUDUDD, UUU*DDD, where U=(1,1), D=(1,-1) and * indicates a peak at odd level. Number of ascents of length 1 in all Dyck paths of semilength n+1 (an ascent in a Dyck path is a maximal string of up steps). Example: a(2)=6 because we have uDuDuD, uDUUDD, UUDDuD, UUDuDD, UUUDDD, where an ascent of length 1 is indicated by a lower case letter. - _Emeric Deutsch_, Dec 05 2003

%C a(n-1) = number of subsets of 2n-1 distinct elements taken n at a time that contain a given element. E.g., n=4 -> a(3)=20 and if we consider the subsets of 7 taken 4 at a time with a 1 we get (1234, 1235, 1236, 1237, 1245, 1246, 1247, 1256, 1257, 1267, 1345, 1346, 1347, 1356, 1357, 1367, 1456, 1457, 1467, 1567) and there are 20 of them. - _Jon Perry_, Jan 20 2004

%C The dimension of a particular (necessarily existent) absolutely universal embedding of the unitary dual polar space DSU(2n,q^2) where q>2. - J. Taylor (jt_cpp(AT)yahoo.com), Apr 02 2004.

%C Number of standard tableaux of shape (n+1, 1^n). - _Emeric Deutsch_, May 13 2004

%C Erdős, Graham et al. conjectured that a(n) is never squarefree for sufficiently large n (cf. Graham, Knuth, Patashnik, Concrete Math., 2nd ed., Exercise 112). Sárközy showed that if s(n) is the square part of a(n), then s(n) is asymptotically (sqrt(2)-2)*(sqrt(n))*(Riemann Zeta Function(1/2)). Granville and Ramare proved that the only squarefree values are a(1)=2, a(2)=6 and a(4)=70. - _Jonathan Vos Post_, Dec 04 2004 [For more about this conjecture, see A261009. - _N. J. A. Sloane_, Oct 25 2015]

%C The MathOverflow link contains the following comment (slightly edited): The Erdős squarefree conjecture (that a(n) is never squarefree for n>4) was proved in 1980 by Sárközy, A. (On divisors of binomial coefficients. I. J. Number Theory 20 (1985), no. 1, 70-80.) who showed that the conjecture holds for all sufficiently large values of n, and by A. Granville and O. Ramaré (Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika 43 (1996), no. 1, 73-107) who showed that it holds for all n>4. - Fedor Petrov, Nov 13 2010. [From _N. J. A. Sloane_, Oct 29 2015]

%C p divides a((p-1)/2)-1=A030662(n) for prime p=5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, ... = A002144(n) Pythagorean primes: primes of form 4n+1. - _Alexander Adamchuk_, Jul 04 2006

%C The number of direct routes from my home to Granny's when Granny lives n blocks south and n blocks east of my home in Grid City. To obtain a direct route, from the 2n blocks, choose n blocks on which one travels south. For example, a(2)=6 because there are 6 direct routes: SSEE, SESE, SEES, EESS, ESES and ESSE. - _Dennis P. Walsh_, Oct 27 2006

%C Inverse: With q = -log(log(16)/(pi a(n)^2)), ceiling((q + log(q))/log(16)) = n. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 26 2007

%C Number of partitions with Ferrers diagrams that fit in an n X n box (including the empty partition of 0). Example: a(2) = 6 because we have: empty, 1, 2, 11, 21 and 22. - _Emeric Deutsch_, Oct 02 2007

%C So this is the 2-dimensional analog of A008793. - _William Entriken_, Aug 06 2013

%C The number of walks of length 2n on an infinite linear lattice that begins and ends at the origin. - Stefan Hollos (stefan(AT)exstrom.com), Dec 10 2007

%C The number of lattice paths from (0,0) to (n,n) using steps (1,0) and (0,1). - _Joerg Arndt_, Jul 01 2011

%C Integral representation: C(2n,n)=1/Pi Integral [(2x)^(2n)/sqrt(1 - x^2),{x,-1, 1}], i.e., C(2n,n)/4^n is the moment of order 2n of the arcsin distribution on the interval (-1,1). - _N-E. Fahssi_, Jan 02 2008

%C Also the Catalan transform of A000079. - _R. J. Mathar_, Nov 06 2008

%C Straub, Amdeberhan and Moll: "... it is conjectured that there are only finitely many indices n such that C_n is not divisible by any of 3, 5, 7 and 11." - _Jonathan Vos Post_, Nov 14 2008

%C Equals INVERT transform of A081696: (1, 1, 3, 9, 29, 97, 333, ...). - _Gary W. Adamson_, May 15 2009

%C Also, in sports, the number of ordered ways for a "Best of 2n-1 Series" to progress. For example, a(2) = 6 means there are six ordered ways for a "best of 3" series to progress. If we write A for a win by "team A" and B for a win by "team B" and if we list the played games chronologically from left to right then the six ways are AA, ABA, BAA, BB, BAB, and ABB. (Proof: To generate the a(n) ordered ways: Write down all a(n) ways to designate n of 2n games as won by team A. Remove the maximal suffix of identical letters from each of these.) - _Lee A. Newberg_, Jun 02 2009

%C Number of n X n binary arrays with rows, considered as binary numbers, in nondecreasing order, and columns, considered as binary numbers, in nonincreasing order. - _R. H. Hardin_, Jun 27 2009

%C Hankel transform is 2^n. - _Paul Barry_, Aug 05 2009

%C It appears that a(n) is also the number of quivers in the mutation class of twisted type BC_n for n>=2.

%C Central terms of Pascal's triangle: a(n) = A007318(2*n,n). - _Reinhard Zumkeller_, Nov 09 2011

%C Number of words on {a,b} of length 2n such that no prefix of the word contains more b's than a's. - _Jonathan Nilsson_, Apr 18 2012

%C From Pascal's triangle take row(n) with terms in order a1,a2,..a(n) and row(n+1) with terms b1,b2,..b(n), then 2*(a1*b1 + a2*b2 + ... + a(n)*b(n)) to get the terms in this sequence. - _J. M. Bergot_, Oct 07 2012. For example using rows 4 and 5: 2*(1*(1) + 4*(5) + 6*(10) + 4*(10) + 1*(5)) = 252, the sixth term in this sequence.

%C Take from Pascal's triangle row(n) with terms b1, b2, ..., b(n+1) and row(n+2) with terms c1, c2, ..., c(n+3) and find the sum b1*c2 + b2*c3 + ... + b(n+1)*c(n+2) to get A000984(n+1).  Example using row(3) and row(5) gives sum 1*(5)+3*(10)+3*(10)+1*(5) = 70 = A000984(4). - _J. M. Bergot_, Oct 31 2012

%C a(n) == 2 mod n^3 iff n is a prime > 3. (See Mestrovic link, p. 4.) - _Gary Detlefs_, Feb 16 2013

%C Conjecture: For any positive integer n, the polynomial sum_{k=0}^n a(k)x^k is irreducible over the field of rational numbers. In general, for any integer m>1 and n>0, the polynomial f_{m,n}(x) = Sum_{k=0..n} (m*k)!/(k!)^m*x^k is irreducible over the field of rational numbers. - _Zhi-Wei Sun_, Mar 23 2013

%C This comment generalizes the comment dated Oct 31 2012 and the second of the sequence's original comments.  For j = 1 to n, a(n) = Sum_{k=0..j} C(j,k)* C(2n-j, n-k) = 2*Sum_{k=0..j-1} C(j-1,k)*C(2n-j, n-k). - _Charlie Marion_, Jun 07 2013

%C The differences between consecutive terms of the sequence of the quotients between consecutive terms of this sequence form a sequence containing the reciprocals of the triangular numbers. In other words, a(n+1)/a(n)-a(n)/a(n-1) = 2/(n*(n+1)). - _Christian Schulz_, Jun 08 2013

%C Number of distinct strings of length 2n using n letters A and n letters B. - _Hans Havermann_, May 07 2014

%C From _Fung Lam_, May 19 2014: (Start)

%C Expansion of  G.f. A(x) = 1/(1+q*x*c(x)), where parameter q is positive or negative (except q=-1), and c(x) is the g.f. of A000108 for Catalan numbers. The case of q=-1 recovers the g.f. of A000108 as xA^2-A+1=0. The present sequence A000984 refers to q=-2. Recurrence: (1+q)*(n+2)*a(n+2) + ((q*q-4*q-4)*n + 2*(q*q-q-1))*a(n+1) - 2*q*q*(2*n+1)*a(n) = 0, a(0)=1, a(1)=-q. Asymptotics: a(n) ~ ((q+2)/(q+1))*(q^2/(-q-1))^n, q<=-3, a(n) ~ (-1)^n*((q+2)/(q+1))*(q^2/(q+1))^n, q>=5, and a(n) ~ -Kq*2^(2*n)/sqrt(Pi*n^3), where the multiplicative constant Kq is given by K1=1/9 (q=1), K2=1/8 (q=2), K3=3/25 (q=3), K4=1/9 (q=4). These formulas apply to existing sequences A126983 (q=1), A126984 (q=2), A126982 (q=3), A126986 (q=4), A126987 (q=5), A127017 (q=6), A127016 (q=7), A126985 (q=8), A127053 (q=9), and to A007854 (q=-3), A076035 (q=-4), A076036 (q=-5), A127628 (q=-6), A126694 (q=-7), A115970 (q=-8). (End)

%C a(n)*(2^n)^(j-2) equals S(n), where S(n) is the n-th number in the self-convolved sequence which yields the powers of 2^j for all integers j, n>=0. For example, when n=5 and j=4, a(5)=252; 252*(2^5)^(4-2) = 252*1024 = 258048.  The self-convolved sequence which yields powers of 16 is {1, 8, 96, 1280, 17920, 258048, ...}; i.e., S(5) = 258048. Note that the convolved sequences will be composed of numbers decreasing from 1 to 0, when j<2 (exception being j=1, where the first two numbers in the sequence are 1 and all others decreasing). - _Bob Selcoe_, Jul 16 2014

%C The variance of the n-th difference of a sequence of pairwise uncorrelated random variables each with variance 1. - _Liam Patrick Roche_, Jun 04 2015

%C Number of ordered trees with n edges where vertices at level 1 can be of 2 colors. Indeed, the standard decomposition of ordered trees leading to the equation C = 1 + zC^2 (C is the Catalan function), yields this time G = 1 + 2zCG, from where G = 1/sqrt(1-4z). - _Emeric Deutsch_, Jun 17 2015

%C Number of monomials of degree at most n in n variables. - _Ran Pan_, Sep 26 2015

%C Let V(n, r) denote the volume of an n-dimensional sphere with radius r, then V(n, 2^n) / Pi = V(n-1, 2^n) * a(n/2) for all even n. - _Peter Luschny_, Oct 12 2015

%C a(n) is the number of sets {i1,...,in} of length n such that n >= i1 >= i2 >= ... >= in >= 0. For instance, a(2) = 6 as there are only 6 such sets: (2,2) (2,1) (2,0) (1,1) (1,0) (0,0). - _Anton Zakharov_, Jul 04 2016

%C From _Ralf Steiner_, Apr 07 2017: (Start)

%C By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:

%C Sum_{k>=0} a(k)/(-2)^k = 1/sqrt(3).

%C Sum_{k>=0} a(k)/(-1)^k = 1/sqrt(5).

%C Sum_{k>=0} a(k)/(-1/2)^k = 1/3.

%C Sum_{k>=0} a(k)/(1/2)^k = -1/sqrt(7)i.

%C Sum_{k>=0} a(k)/(1)^k = -1/sqrt(3)i.

%C Sum_{k>=0} a(k)/2^k  = -i. (End)

%C Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) > e(j). [Martinez and Savage, 2.18] - _Eric M. Schmidt_, Jul 17 2017

%C The o.g.f. for the sequence equals the diagonal of any of the following the rational functions: 1/(1 - (x + y)), 1/(1 - (x + y*z)), 1/(1 - (x + x*y + y*z)) or 1/(1 - (x + y + y*z)). - _Peter Bala_, Jan 30 2018

%C From _Colin Defant_, Sep 16 2018: (Start)

%C Let s denote West's stack-sorting map. a(n) is the number of permutations pi of [n+1] such that s(pi) avoids the patterns 132, 231, and 321. a(n) is also the number of permutations pi of [n+1] such that s(pi) avoids the patterns 132, 312, and 321.

%C a(n) is the number of permutations of [n+1] that avoid the patterns 1342, 3142, 3412, and 3421. (End)

%C All binary self-dual codes of length 4n, for n>0, must contain at least a(n) codewords of weight 2n.  More to the point, there will always be at least one, perhaps unique, binary self-dual code of length 4n that will contain exactly a(n) codewords that have a hamming weight equal to half the length of the code (2n). This code can be constructed by direct summing the unique binary self-dual code of length 2 (up to permutation equivalence) to itself an even number of times. A permutation equivalent code can be constructed by augmenting two identity matrices of length 2n together. - _Nathan J. Russell_, Nov 25 2018

%C From _Isaac Saffold_, Dec 28 2018: (Start)

%C Let [b/p] denote the Legendre symbol and 1/b denote the inverse of b mod p. Then, for m and n, where n is not divisible by p,

%C [(m+n)/p] == [n/p]*Sum_{k=0..(p-1)/2} (-m/(4*n))^k * a(k) (mod p).

%C Evaluating this identity for m = -1 and n = 1 demonstrates that, for all odd primes p, Sum_{k=0..(p-1)/2} (1/4)^k * a(k) is divisible by p. (End)

%C Number of vertices of the subgraph of the (2n-1)-dimensional hypercube induced by all bitstrings with n-1 or n many 1s. The middle levels conjecture asserts that this graph has a Hamilton cycle. - _Torsten Muetze_, Feb 11 2019

%C a(n) is the number of walks of length 2n from the origin with steps (1,1) and (1,-1) that stay on or above the x-axis. Equivalently, a(n) is the number of walks of length 2n from the origin with steps (1,0) and (0,1) that stay in the first octant. - _Alexander Burstein_, Dec 24 2019

%C Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {3>1, 1>2} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second element but smaller than the third elements. - _Sergey Kitaev_, Dec 08 2020

%C From _Gus Wiseman_, Jul 21 2021: (Start)

%C Also the number of integer compositions of 2n+1 with alternating sum 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(0) = 1 through a(2) = 6 compositions are:

%C   (1)  (2,1)    (3,2)

%C        (1,1,1)  (1,2,2)

%C                 (2,2,1)

%C                 (1,1,2,1)

%C                 (2,1,1,1)

%C                 (1,1,1,1,1)

%C The following relate to these compositions:

%C - The unordered version is A000070.

%C - The alternating sum -1 version is counted by A001791, ranked by A345910/A345912.

%C - The alternating sum 0 version is counted by A088218, ranked by A344619.

%C - Including even indices gives A126869.

%C - The complement is counted by A202736.

%C - Ranked by A345909 (reverse: A345911).

%C Equivalently, a(n) counts binary numbers with 2n+1 digits and one more 1 than 0's. For example, the a(2) = 6 binary numbers are: 10011, 10101, 10110, 11001, 11010, 11100.

%C (End)

%C From _Michael Wallner_, Jan 25 2022: (Start)

%C a(n) is the number of nx2 Young tableaux with a single horizontal wall between the first and second column. If there is a wall between two cells, the entries may be decreasing; see [Banderier, Wallner 2021].

%C   Example for a(2)=6:

%C    3 4  2 4  3 4  3|4  4|3  2|4

%C    1|2, 1|3, 2|1, 1 2, 1 2, 1 3

%C a(n) is also the number of nx2 Young tableaux with n "walls" between the first and second column.

%C   Example for a(2)=6:

%C    3|4  2|4  4|3  3|4  4|3  4|2

%C    1|2, 1|3, 1|2, 2|1, 2|1, 3|1 (End)

%C From _Shel Kaphan_, Jan 12 2023: (Start)

%C a(n)/4^n is the probability that a fair coin tossed 2n times will come up heads exactly n times and tails exactly n times, or that a random walk with steps of +-1 will return to the starting point after 2n steps (not necessarily for the first time). As n becomes large, this number asymptotically approaches 1/sqrt(n*Pi), using Stirling's approximation for n!.

%C a(n)/(4^n*(2n-1)) is the probability that a random walk with steps of +-1 will return to the starting point for the first time after 2n steps. The absolute value of the n-th term of A144704 is denominator of this fraction.

%C Considering all possible random walks of exactly 2n steps with steps of +-1, a(n)/(2n-1) is the number of such walks that return to the starting point for the first time after 2n steps. See the absolute values of A002420 or A284016 for these numbers. For comparison, as mentioned by _Stefan Hollos_, Dec 10 2007, a(n) is the number of such walks that return to the starting point after 2n steps, but not necessarily for the first time. (End)

%C p divides a((p-1)/2) + 1 for primes p of the form 4*k+3 (A002145). - _Jules Beauchamp_, Feb 11 2023

%C Also the size of the shuffle product of two words of length n, such that the union of the two words consist of 2n distinct elements. - _Robert C. Lyons_, Mar 15 2023

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 160.

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575, line -3, with a=b=n.

%D E. Deutsch and L. Shapiro, Seventeen Catalan identities, Bulletin of the Institute of Combinatorics and its Applications, 31, 31-38, 2001.

%D H. W. Gould, Combinatorial Identities, Morgantown, 1972, (3.66), page 30.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, Second Ed., see Exercise 112.

%D M. Griffiths, The Backbone of Pascal's Triangle, United Kingdom Mathematics Trust (2008), 3-124.

%D Leonard Lipshitz and A. van der Poorten. "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990): 339-358.

%D J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe and Edward Jiang, <a href="/A000984/b000984.txt">Table of n, a(n) for n = 0..500</a> (Previously 0..200 by T. D. Noe)

%H J. Abate and W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Whitt/whitt6.html">Brownian Motion and the Generalized Catalan Numbers</a>, J. Int. Seq.  14 (2011) # 11.2.6, example section 3.

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abrate, S. Barbero, U. Cerruti, and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barbero/barbero9.html"> Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators</a>, J. Int. Seq. 14 (2011) # 11.8.1.

%H B. Adamczewski, J. P. Bell, and E. Delaygue, <a href="http://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences "a la Lucas"</a>, arXiv preprint arXiv:1603.04187 [math.NT], 2016.

%H M. Aigner, <a href="http://dx.doi.org/10.1016/j.disc.2007.06.012">Enumeration via ballot numbers</a>, Discrete Math., 308 (2008), 2544-2563.

%H Michael Anshelevich, <a href="https://arxiv.org/abs/1708.08034">Product formulas on posets, Wick products, and a correction for the q-Poisson process</a>, arXiv:1708.08034 [math.OA], 2017, See Proposition 34 p. 25.

%H D. H. Bailey, J. M. Borwein and D. M. Bradley, <a href="http://arXiv.org/abs/math.CA/0505270">Experimental determination of Apéry-like identities for zeta(4n+2)</a>, arXiv:math/0505124 [math.CA], 2005.

%H Cyril Banderier and Michael Wallner, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/47.html">Young Tableaux with Periodic Walls: Counting with the Density Method</a>, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry4/bern2.html">Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences</a>, Journal of Integer Sequences, Vol. 15 2012, #12.8.2.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry1/barry242.html">On the Central Coefficients of Riordan Matrices</a>, Journal of Integer Sequences, 16 (2013), #13.5.1.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.html">A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.4.

%H Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry1/barry411.html">The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths</a>, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.

%H Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Barry/barry444.html">On the Central Antecedents of Integer (and Other) Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.

%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.

%H Paul Barry, <a href="http://dx.doi.org/10.1155/2013/657806">On the Connection Coefficients of the Chebyshev-Boubaker polynomials</a>, The Scientific World Journal, Volume 2013 (2013), Article ID 657806, 10 pages.

%H A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rinaldi/rinaldi5.html">Permutations defining convex permutominoes</a>, J. Int. Seq. 10 (2007) # 07.9.7.

%H Robert J. Betts, <a href="http://arxiv.org/abs/1010.3070">Lack of Divisibility of {2N choose N} by three fixed odd primes infinitely often, through the Extension of a Result by P. Erdős, et al.</a>, arXiv:1010.3070 [math.NT], 2010. [It is not clear if the results in this paper have been confirmed. There appears to be no mention of this work in MathSciNet, for example. - _N. J. A. Sloane_, Oct 29 2015]

%H J. Borwein and D. Bradley, <a href="https://arxiv.org/abs/math/0505124">Empirically determined Apéry-like formulas for zeta(4n+3)</a>, arXiv:math/0505124 [math.CA], 2005.

%H Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, <a href="http://www.carmamaths.org/resources/jon/walks.pdf">Random Walk Integrals</a>, 2010.

%H Jonathan M. Borwein and Armin Straub, <a href="http://carmamaths.org/resources/jon/wmi-paper.pdf">Mahler measures, short walks and log-sine integrals</a>.

%H H. J. Brothers, <a href="http://www.brotherstechnology.com/docs/Pascal&#39;s_Prism_(supplement).pdf">Pascal's Prism: Supplementary Material</a>.

%H Marie-Louise Bruner, <a href="http://arxiv.org/abs/1505.04929">Central binomial coefficients also count (2431,4231,1432,4132)-avoiders</a>, arXiv:1505.04929 [math.CO], 2015.

%H Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, <a href="https://arxiv.org/abs/1903.01095">The Number of Convex Polyominoes with Given Height and Width</a>, arXiv:1903.01095 [math.CO], 2019.

%H N. T. Cameron, <a href="https://www.math.hmc.edu/~cameron/dissertation.pdf">Random walks, trees and extensions of Riordan group techniques</a>, Dissertation, Howard University, 2002.

%H G. Chatel and V. Pilaud, <a href="http://arxiv.org/abs/1411.3704">The Cambrian and Baxter-Cambrian Hopf Algebras</a>, arXiv preprint arXiv:1411.3704 [math.CO], 2014-2015.

%H Hongwei Chen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Chen/chen78.html">Evaluations of Some Variant Euler Sums</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.

%H G.-S. Cheon, H. Kim, and L. W. Shapiro, <a href="http://arxiv.org/abs/1410.1249">Mutation effects in ordered trees</a>, arXiv preprint arXiv:1410.1249 [math.CO], 2014.

%H J. Cigler, <a href="http://arxiv.org/abs/1109.1449">Some nice Hankel determinants</a>, arXiv:1109.1449 [math.CO], 2011.

%H Johann Cigler and Christian Krattenthaler, <a href="https://arxiv.org/abs/2003.01676">Hankel determinants of linear combinations of moments of orthogonal polynomials</a>, arXiv:2003.01676 [math.CO], 2020.

%H CombOS - Combinatorial Object Server, <a href="http://combos.org/middle.html">Generate middle levels Gray codes</a>

%H B. N. Cooperstein and E. E. Shult, <a href="http://www.emis.de/journals/AG/1-1/1_037.pdf">A note on embedding and generating dual polar spaces</a>. Adv. Geom. 1 (2001), 37-48. See Theorem 5.4.

%H Kristina Crona, Mengming Luo, and Devin Greene, <a href="https://doi.org/10.1016/j.jtbi.2020.110155">An Uncertainty Law for Microbial Evolution</a>, Journal of Theoretical Biology (2020) Vol. 489, Article No. 110155.

%H D. Daly and L. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/sandiego2013.pdf">Pattern avoidance in rook monoids</a>, 2013.

%H Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.

%H Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Davenport/dav3.html">The Boundary of Ordered Trees</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.

%H Thierry Dana-Picard, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Dana-Picard/dana23.html">Sequences of Definite Integrals, Factorials and Double Factorials</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.6.

%H Isaac DeJager, Madeleine Naquin, and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.

%H E. Delaygue, <a href="http://arxiv.org/abs/1310.4131">Arithmetic properties of Apéry-like numbers</a>, arXiv preprint arXiv:1310.4131 [math.NT], 2013.

%H Nachum Dershowitz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Dershowitz/dersh3.html">Touchard's Drunkard</a>, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

%H Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.disc.2003.10.014">Enumerating symmetric directed convex polyominoes</a>, Discrete Math., 280 (2004), 225-231.

%H Satyan L. Devadoss, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.092">A realization of graph associahedra</a>, Discrete Math. 309 (2009), no. 1, 271-276.

%H J. C. F. de Winter, <a href="http://pareonline.net/getvn.asp?v=18&amp;n=10">Using the Student's t-test with extremely small sample sizes</a>, Practical Assessment, Research & Evaluation, 18(10), 2013.

%H R. M. Dickau, <a href="https://web.archive.org/web/20200710012924/http://mathforum.org:80/advanced/robertd/manhattan.html">Shortest-path diagrams</a>

%H Phan Thuan Do, Thi Thu Huong Tran, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/1809.00742">Exhaustive generation for permutations avoiding a (colored) regular sets of patterns</a>, arXiv:1809.00742 [cs.DM], 2018.

%H R. Duarte and A. G. de Oliveira, <a href="http://arxiv.org/abs/1302.2100">Short note on the convolution of binomial coefficients</a>, arXiv preprint arXiv:1302.2100 [math.CO], 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Duarte/duarte3.html">J. Int. Seq. 16 (2013) #13.7.6</a>.

%H Bryan Ek, <a href="https://arxiv.org/abs/1804.05933">Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics</a>, arXiv:1804.05933 [math.CO], 2018.

%H P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0369288-3">On the prime factors of C(2n,n)</a>, Math. Comp. 29 (1975), 83-92.

%H Gennady Eremin, <a href="https://arxiv.org/abs/2003.01494"> Factoring Middle Binomial Coefficients</a>, arXiv:2003.01494 [math.CO], 2020.

%H A. Erickson and F. Ruskey, <a href="http://arxiv.org/abs/1304.0070">Enumerating maximal tatami mat coverings of square grids with v vertical dominoes</a>, arXiv preprint arXiv:1304.0070 [math.CO], 2013.

%H Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">J. Int. Seq. 17 (2014) #14.1.5</a>.

%H Francesc Fité and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1203.1476">Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1</a>, arXiv preprint arXiv:1203.1476 [math.NT], 2012.

%H Francesc Fité, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv:1110.6638 [math.NT], 2011.

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 77.

%H Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.

%H Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.

%H Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.

%H H. W. Gould, <a href="http://www.math.wvu.edu/~gould/">Tables of Combinatorial Identities, Vol. 7</a>, Edited by J. Quaintance.

%H A. Granville and O. Ramaré, <a href="http://www.dms.umontreal.ca/~andrew/PDF/ramare.pdf">Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients</a>, Mathematika 43 (1996), 73-107, <a href="http://dx.doi.org/10.1112/S0025579300011608">[DOI]</a>.

%H T. Halverson and M. Reeks, <a href="http://arxiv.org/abs/1302.6150">Gelfand Models for Diagram Algebras</a>, arXiv preprint arXiv:1302.6150 [math.RT], 2013.

%H Oktay Haracci (timetunnel3(AT)hotmail.com), <a href="https://web.archive.org/web/20091027100800/http://www.geocities.com/timeparadox/ismi_azam.html">Regular Polygons</a>

%H R. H. Hardin, <a href="/A151801/a151801.txt">Binary arrays with both rows and cols sorted, symmetries</a>

%H P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, <a href="https://doi.org/10.37236/3693">Congruences of Finite Summations of the Coefficients in certain Generating Functions</a>, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.

%H W. Cary Huffman and Vera Pless, <a href="https://doi.org/10.1017/CBO9780511807077">Fundamentals of Error Correcting Codes</a>, Cambridge University Press, 2003, Pages 7,252-282,338-393.

%H Anders Hyllengren, <a href="/A001006/a001006_5.pdf">Four integer sequences</a>, Oct 04 1985. Observes essentially that A000984 and A002426 are inverse binomial transforms of each other, as are A000108 and A001006.

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H I. Jensen, <a href="https://researchers.ms.unimelb.edu.au/~ij@unimelb/polygons/Polygons_ser.html">Series expansions for self-avoiding polygons</a>

%H Pakawut Jiradilok and Elchanan Mossel, <a href="https://arxiv.org/abs/2402.11990">Gaussian Broadcast on Grids</a>, arXiv:2402.11990 [cs.IT], 2024. See p. 27.

%H C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Kimberling/kimberling24.html">Matrix Transformations of Integer Sequences</a>, J. Integer Seqs., Vol. 6, 2003.

%H Sergey Kitaev and Jeffrey Remmel, <a href="http://arxiv.org/abs/1201.1323">Simple marked mesh patterns</a>, arXiv preprint arXiv:1201.1323 [math.CO], 2012.

%H V. V. Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1206.0877">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, arXiv:1206.0877 [math.CO], 2012.

%H D. Kruchinin and V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, Journal of Integer Sequence,  Vol. 15 (2012), article 12.9.3.

%H Jean-Philippe Labbé and Carsten Lange, <a href="https://arxiv.org/abs/1802.07978">Cambrian acyclic domains: counting c-singletons</a>, arXiv:1802.07978 [math.CO], 2018.

%H Marie-Louise Lackner and M. Wallner, <a href="http://dmg.tuwien.ac.at/mwallner/files/lpintro.pdf">An invitation to analytic combinatorics and lattice path counting</a>; Preprint, Dec 2015.

%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)

%H J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2322496">Interesting series involving the Central Binomial Coefficient</a>, Am. Math. Monthly 92, no 7 (1985) 449-457.

%H Huyile Liang, Jeffrey Remmel, and Sainan Zheng, <a href="https://arxiv.org/abs/1710.05795">Stieltjes moment sequences of polynomials</a>, arXiv:1710.05795 [math.CO], 2017, see page 19.

%H L. Lipshitz and A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/a084.pdf">Rational functions, diagonals, automata and arithmetic</a>

%H T. Manneville and V. Pilaud, <a href="http://arxiv.org/abs/1501.07152">Compatibility fans for graphical nested complexes</a>, arXiv preprint arXiv:1501.07152 [math.CO], 2015.

%H Megan A. Martinez and Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016.

%H MathOverflow, <a href="http://mathoverflow.net/questions/45923/divisibility-of-a-binomial-coefficient-by-p2-current-status">Divisibility of a binomial coefficient by p^2 — current status</a>

%H R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv preprint arXiv:1111.3057 [math.NT], 2011.

%H R. Mestrovic, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.

%H W. Mlotkowski and K. A. Penson, <a href="http://arxiv.org/abs/1309.0595">Probability distributions with binomial moments</a>, arXiv preprint arXiv:1309.0595 [math.PR], 2013.

%H T. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1945-08486-9">The hypersurface cross ratio</a>, Bull. Amer. Math. Soc., 51 (1945), 976-984.

%H T. S. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1948-09002-4 ">Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products</a>, Bull. Amer. Math. Soc., 54 (1948), 352-360.

%H Torsten Mütze, <a href="http://arxiv.org/abs/1404.4442">Proof of the middle levels conjecture</a>, arXiv preprint arXiv:1404.4442 [math.CO], 2014.

%H Asamoah Nkwanta and Earl R. Barnes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Nkwanta/nkwanta2.html">Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind</a>, Journal of Integer Sequences, Article 12.3.3, 2012. - From _N. J. A. Sloane_, Sep 16 2012

%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

%H Valentin Ovsienko, <a href="https://arxiv.org/abs/2111.02553">Shadow sequences of integers, from Fibonacci to Markov and back</a>, arXiv:2111.02553 [math.CO], 2021.

%H Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/warmups/eI.html">Exercise I</a>, Project P.

%H P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/PEART/peart1.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.

%H M. A. Perlstadt, <a href="http://dx.doi.org/10.1016/0022-314X(87)90069-2">Some Recurrences for Sums of Powers of Binomial Coefficients</a>, Journal of Number Theory 27 (1987), pp. 304-309.

%H A. Petojevic and N. Dapic, <a href="http://www.mi.sanu.ac.rs/~gvm/radovi/AP-Budva.pdf">The vAm(a,b,c;z) function</a>, Preprint 2013.

%H C. Pomerance, <a href="https://math.dartmouth.edu/~carlp/catalan5.pdf">Divisors of the middle binomial coefficient</a>, Amer. Math. Monthly, 112 (2015), 636-644.

%H Y. Puri and T. Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.

%H T. M. Richardson, <a href="http://arxiv.org/abs/1405.6315">The Reciprocal Pascal Matrix</a>, arXiv preprint arXiv:1405.6315 [math.CO], 2014.

%H John Riordan, <a href="/A002720/a002720_3.pdf">Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences</a>. Note that the sequences are identified by their N-numbers, not their A-numbers.

%H D. P. Roberts and A. Venkatesh, <a href="http://math.stanford.edu/~akshay/research/full.pdf">Hurwitz monodromy and full number fields</a>, 2014. Also arXiv:1401:7379, 2014.

%H H. P. Robinson, <a href="/A006530/a006530.pdf">Letter to N. J. A. Sloane, Oct 1981</a>

%H A. Sárközy, <a href="http://dx.doi.org/10.1016/0022-314X(85)90017-4">On Divisors of Binomial Coefficients. I.</a>, J. Number Th. 20, 70-80, 1985.

%H J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].

%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)

%H L. W. Shapiro, S. Getu, Wen-Jin Woan and L. C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan Group</a>, Discrete Appl. Maths. 34 (1991) 229-239.

%H N. J. A. Sloane, <a href="/A000984/a000984.pdf">Notes on A984 and A2420-A2424</a>

%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

%H Armin Straub, <a href="http://arminstraub.com/pub/dissertation">Arithmetic aspects of random walks and methods in definite integration</a>, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012.

%H Armin Straub, Tewodros Amdeberhan and Victor H. Moll, <a href="http://arxiv.org/abs/0811.2028">The p-adic valuation of k-central binomial coefficients</a>, arXiv:0811.2028 [math.NT], 2008, pp. 10-11.

%H V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.

%H R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SULANKE/sulanke.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.

%H Hua Sun and Yi Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Wang/wang11.html">A Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers</a>, J. Int. Seq. 17 (2014) # 14.5.2.

%H Michael Torpey, <a href="https://doi.org/10.17630/10023-17350">Semigroup congruences: computational techniques and theoretical applications</a>, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).

%H H. A. Verrill, <a href="https://arxiv.org/abs/math/0407327">Sums of squares of binomial coefficients, ...</a>, arXiv:math/0407327 [math.CO], 2004.

%H M. Wallner, <a href="http://dmg.tuwien.ac.at/drmota/Thesis_Wallner.pdf">Lattice Path Combinatorics</a>, Diplomarbeit, Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien, 2013.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StaircaseWalk.html">Staircase Walk</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CircleLinePicking.html">Circle Line Picking</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Shuffle_algebra#Shuffle_product">Shuffle product</a>.

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F a(n)/(n+1) = A000108(n), the Catalan numbers.

%F G.f.: A(x) = (1 - 4*x)^(-1/2) = 1F0(1/2;;4x).

%F a(n+1) = 2*A001700(n) = A030662(n) + 1. a(2*n) = A001448(n), a(2*n+1) = 2*A002458(n) =A099976.

%F D-finite with recurrence: n*a(n) + 2*(1-2*n)*a(n-1)=0.

%F a(n) = 2^n/n! * Product_{k=0..n-1} (2*k+1).

%F a(n) = a(n-1)*(4-2/n) = Product_{k=1..n} (4-2/k) = 4*a(n-1) + A002420(n) = A000142(2*n)/(A000142(n)^2) = A001813(n)/A000142(n) = sqrt(A002894(n)) = A010050(n)/A001044(n) = (n+1)*A000108(n) = -A005408(n-1)*A002420(n). - _Henry Bottomley_, Nov 10 2000

%F Using Stirling's formula in A000142 it is easy to get the asymptotic expression a(n) ~ 4^n / sqrt(Pi * n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

%F Integral representation as n-th moment of a positive function on the interval [0, 4]: a(n) =  Integral_{x=0..4}(x^n*((x*(4-x))^(-1/2))/Pi), n=0, 1, ... This representation is unique. - _Karol A. Penson_, Sep 17 2001

%F Sum_{n>=1} 1/a(n) = (2*Pi*sqrt(3) + 9)/27. [Lehmer 1985, eq. (15)] - _Benoit Cloitre_, May 01 2002 (= A073016. - _Bernard Schott_, Jul 20 2022)

%F a(n) = Max_{ (i+j)!/(i!j!) | 0<=i,j<=n }. - _Benoit Cloitre_, May 30 2002

%F a(n) = Sum_{k=0..n} binomial(n+k-1,k), row sums of A059481. - _Vladeta Jovovic_, Aug 28 2002

%F E.g.f.: exp(2*x)*I_0(2x), where I_0 is Bessel function. - _Michael Somos_, Sep 08 2002

%F E.g.f.: I_0(2*x) = Sum a(n)*x^(2*n)/(2*n)!, where I_0 is Bessel function. - _Michael Somos_, Sep 09 2002

%F a(n) = Sum_{k=0..n} binomial(n, k)^2. - _Benoit Cloitre_, Jan 31 2003

%F Determinant of n X n matrix M(i, j) = binomial(n+i, j). - _Benoit Cloitre_, Aug 28 2003

%F Given m = C(2*n, n), let f be the inverse function, so that f(m) = n. Letting q denote -log(log(16)/(m^2*Pi)), we have f(m) = ceiling( (q + log(q)) / log(16) ). - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Oct 30 2003

%F a(n) = 2*Sum_{k=0..(n-1)} a(k)*a(n-k+1)/(k+1). - _Philippe Deléham_, Jan 01 2004

%F a(n+1) = Sum_{j=n..n*2+1} binomial(j, n). E.g., a(4) = C(7,3) + C(6,3) + C(5,3) + C(4,3) + C(3,3) = 35 + 20 + 10 + 4 + 1 = 70. - _Jon Perry_, Jan 20 2004

%F a(n) = (-1)^(n)*Sum_{j=0..(2*n)} (-1)^j*binomial(2*n, j)^2. - Helena Verrill (verrill(AT)math.lsu.edu), Jul 12 2004

%F a(n) = Sum_{k=0..n} binomial(2n+1, k)*sin((2n-2k+1)*Pi/2). - _Paul Barry_, Nov 02 2004

%F a(n-1) = (1/2)*(-1)^n*Sum_{0<=i, j<=n}(-1)^(i+j)*binomial(2n, i+j). - _Benoit Cloitre_, Jun 18 2005

%F a(n) = C(2n, n-1) + C(n) = A001791(n) + A000108(n). - _Lekraj Beedassy_, Aug 02 2005

%F G.f.: c(x)^2/(2*c(x)-c(x)^2) where c(x) is the g.f. of A000108. - _Paul Barry_, Feb 03 2006

%F a(n) = A006480(n) / A005809(n). - _Zerinvary Lajos_, Jun 28 2007

%F a(n) = Sum_{k=0..n} A106566(n,k)*2^k. - _Philippe Deléham_, Aug 25 2007

%F a(n) = Sum_{k>=0} A039599(n, k). a(n) = Sum_{k>=0} A050165(n, k). a(n) = Sum_{k>=0} A059365(n, k)*2^k, n>0. a(n+1) = Sum_{k>=0} A009766(n, k)*2^(n-k+1). - _Philippe Deléham_, Jan 01 2004

%F a(n) = 4^n*Sum_{k=0..n} C(n,k)(-4)^(-k)*A000108(n+k). - _Paul Barry_, Oct 18 2007

%F a(n) = Sum_{k=0..n} A039598(n,k)*A059841(k). - _Philippe Deléham_, Nov 12 2008

%F A007814(a(n)) = A000120(n). - _Vladimir Shevelev_, Jul 20 2009

%F From _Paul Barry_, Aug 05 2009: (Start)

%F G.f.: 1/(1-2x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction);

%F G.f.: 1/(1-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)

%F If n>=3 is prime, then a(n) == 2 (mod 2*n). - _Vladimir Shevelev_, Sep 05 2010

%F Let A(x) be the g.f. and B(x) = A(-x), then B(x) = sqrt(1-4*x*B(x)^2). - _Vladimir Kruchinin_, Jan 16 2011

%F a(n) = (-4)^n*sqrt(Pi)/(gamma((1/2-n))*gamma(1+n)). - _Gerry Martens_, May 03 2011

%F a(n) = upper left term in M^n, M = the infinite square production matrix:

%F 2, 2, 0, 0, 0, 0, ...

%F 1, 1, 1, 0, 0, 0, ...

%F 1, 1, 1, 1, 0, 0, ...

%F 1, 1, 1, 1, 1, 0, ...

%F 1, 1, 1, 1, 1, 1, ....

%F - _Gary W. Adamson_, Jul 14 2011

%F a(n) = Hypergeometric([-n,-n],[1],1). - _Peter Luschny_, Nov 01 2011

%F E.g.f.: hypergeometric([1/2],[1],4*x). - _Wolfdieter Lang_, Jan 13 2012

%F a(n) = 2*Sum_{k=0..n-1} a(k)*A000108(n-k-1). - _Alzhekeyev Ascar M_, Mar 09 2012

%F G.f.: 1 + 2*x/(U(0)-2*x) where U(k) = 2*(2*k+1)*x + (k+1) - 2*(k+1)*(2*k+3)*x/U(k+1); (continued fraction, Euler's 1st kind, 1-step). - _Sergei N. Gladkovskii_, Jun 28 2012

%F a(n) = Sum_{k=0..n} binomial(n,k)^2*H(k)/(2*H(n)-H(2*n)), n>0, where H(n) is the n-th harmonic number. - _Gary Detlefs_, Mar 19 2013

%F G.f.: Q(0)*(1-4*x), where Q(k) = 1 + 4*(2*k+1)*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 11 2013

%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013

%F E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)^2/(2*k+1)/E(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 01 2013

%F Special values of Jacobi polynomials, in Maple notation: a(n) = 4^n*JacobiP(n,0,-1/2-n,-1). - _Karol A. Penson_, Jul 27 2013

%F a(n) = 2^(4*n)/((2*n+1)*Sum_{k=0..n} (-1)^k*C(2*n+1,n-k)/(2*k+1)). - _Mircea Merca_, Nov 12 2013

%F a(n) = C(2*n-1,n-1)*C(4*n^2,2)/(3*n*C(2*n+1,3)), n>0. - _Gary Detlefs_, Jan 02 2014

%F Sum_{n>=0} a(n)/n! = A234846. - _Richard R. Forberg_, Feb 10 2014

%F 0 =  a(n)*(16*a(n+1) - 6*a(n+2)) + a(n+1)*(-2*a(n+1) + a(n+2)) for all n in Z. - _Michael Somos_, Sep 17 2014

%F a(n+1) = 4*a(n) - 2*A000108(n). Also a(n) = 4^n*Product_{k=1..n}(1-1/(2*k)). - _Stanislav Sykora_, Aug 09 2014

%F G.f.: Sum_{n>=0} x^n/(1-x)^(2*n+1) * Sum_{k=0..n} C(n,k)^2 * x^k. - _Paul D. Hanna_, Nov 08 2014

%F a(n) = (-4)^n*binomial(-1/2,n). - _Jean-François Alcover_, Feb 10 2015

%F a(n) = 4^n*hypergeom([-n,1/2],[1],1). - _Peter Luschny_, May 19 2015

%F a(n) = Sum_{k=0..floor(n/2)} C(n,k)*C(n-k,k)*2^(n-2*k). - _Robert FERREOL_, Aug 29 2015

%F a(n) ~ 4^n*(2-2/(8*n+2)^2+21/(8*n+2)^4-671/(8*n+2)^6+45081/(8*n+2)^8)/sqrt((4*n+1) *Pi). - _Peter Luschny_, Oct 14 2015

%F A(-x) = 1/x * series reversion( x*(2*x + sqrt(1 + 4*x^2)) ). Compare with the o.g.f. B(x) of A098616, which satisfies B(-x) = 1/x * series reversion( x*(2*x + sqrt(1 - 4*x^2)) ). See also A214377. - _Peter Bala_, Oct 19 2015

%F a(n) = GegenbauerC(n,-n,-1). - _Peter Luschny_, May 07 2016

%F a(n) = gamma(1+2*n)/gamma(1+n)^2. - _Andres Cicuttin_, May 30 2016

%F Sum_{n>=0} (-1)^n/a(n) = 4*(5 - sqrt(5)*log(phi))/25 = 0.6278364236143983844442267..., where phi is the golden ratio. - _Ilya Gutkovskiy_, Jul 04 2016

%F From _Peter Bala_, Jul 22 2016: (Start)

%F This sequence occurs as the closed-form expression for several binomial sums:

%F a(n) = Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n,k)*binomial(2*n + 1,k).

%F a(n) = 2*Sum_{k = 0..2*n-1} (-1)^(n+k)*binomial(2*n - 1,k)*binomial(2*n,k) for n >= 1.

%F a(n) = 2*Sum_{k = 0..n-1} binomial(n - 1,k)*binomial(n,k) for n >= 1.

%F a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)*binomial(x + k,n)*binomial(y + k,n) = Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)*binomial(x - k,n)*binomial(y - k,n) for arbitrary x and y.

%F For m = 3,4,5,... both Sum_{k = 0..m*n} (-1)^k*binomial(m*n,k)*binomial(x + k,n)*binomial(y + k,n) and Sum_{k = 0..m*n} (-1)^k*binomial(m*n,k)*binomial(x - k,n)*binomial(y - k,n) appear to equal Kronecker's delta(n,0).

%F a(n) = (-1)^n*Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)*binomial(x + k,n)*binomial(y - k,n) for arbitrary x and y.

%F For m = 3,4,5,... Sum_{k = 0..m*n} (-1)^k*binomial(m*n,k)*binomial(x + k,n)*binomial(y - k,n) appears to equal Kronecker's delta(n,0).

%F a(n) = Sum_{k = 0..2n} (-1)^k*binomial(2*n,k)*binomial(3*n - k,n)^2 = Sum_{k = 0..2*n} (-1)^k*binomial(2*n,k)* binomial(n + k,n)^2. (Gould, Vol. 7, 5.23).

%F a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(2*n,n + k)*binomial(n + k,n)^2. (End)

%F From _Ralf Steiner_, Apr 07 2017: (Start)

%F Sum_{k>=0} a(k)/(p/q)^k = sqrt(p/(p-4q)) for q in N, p in Z/{-4q< (some p) <-2}.

%F ...

%F Sum_{k>=0} a(k)/(-4)^k = 1/sqrt(2).

%F Sum_{k>=0} a(k)/(17/4)^k = sqrt(17).

%F Sum_{k>=0} a(k)/(18/4)^k = 3.

%F Sum_{k>=0} a(k)/5^k = sqrt(5).

%F Sum_{k>=0} a(k)/6^k = sqrt(3).

%F Sum_{k>=0} a(k)/8^k = sqrt(2).

%F ...

%F Sum_{k>=0} a(k)/(p/q)^k = sqrt(p/(p-4q)) for p>4q.(End)

%F Boas-Buck recurrence: a(n) = (2/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n, 0). See a comment there. - _Wolfdieter Lang_, Aug 10 2017

%F a(n) = Sum_{k = 0..n} (-1)^(n-k) * binomial(2*n+1, k) for n in N. - _Rene Adad_, Sep 30 2017

%F a(n) = A034870(n,n). - _Franck Maminirina Ramaharo_, Nov 26 2018

%F From _Jianing Song_, Apr 10 2022: (Start)

%F G.f. for {1/a(n)}: 4*(sqrt(4-x) + sqrt(x)*arcsin(sqrt(x)/2)) / (4-x)^(3/2).

%F E.g.f. for {1/a(n)}: 1 + exp(x/4)*sqrt(Pi*x)*erf(sqrt(x)/2)/2.

%F Sum_{n>=0} (-1)^n/a(n) = 4*(1/5 - arcsinh(1/2)/(5*sqrt(5))). (End)

%F From _Peter Luschny_, Sep 08 2022: (Start)

%F a(n) = 2^(2*n)*Product_{k=1..2*n} k^((-1)^(k+1)) = A056040(2*n).

%F a(n) = A001316(n) * A356637(n) * A261130(n) for n >= 2. (End)

%F a(n) = 4^n*binomial(n-1/2,-1/2) = 4^n*GegenbauerC(n,1/4,1). - _Gerry Martens_, Oct 19 2022

%F Occurs on the right-hand side of the binomial sum identities Sum_{k = -n..n} (-1)^k * (n + x - k) * binomial(2*n, n+k)^2 = (x + n)*a(n) and Sum_{k = -n..n} (-1)^k * (n + x - k)^2 * binomial(2*n, n+k)^3 = x*(x + 2*n)*a(n) (x arbitrary). Compare with the identity: Sum_{k = -n..n} (-1)^k * binomial(2*n, n+k)^2 = a(n). - _Peter Bala_, Jul 31 2023

%F From _Peter Bala_, Mar 31 2024: (Start)

%F 4^n*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).

%F 16^n = Sum_{k = 0..2*n} a(k)*a(2*n-k). (End)

%F From _Gary Detlefs_, May 28 2024: (Start)

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n,2k)*binomial(2*k,k)*2^(n-2*k). (H. W. Gould) - _Gary Detlefs_, May 28 2024

%F a(n) = Sum_{k=0..2*n} (-1)^k*binomial(2n,k)*binomial(2*n+2*k,n+k)*3^(2*n-k). (H. W. Gould) (End)

%F a(n) = Product_{k>=n+1} k^2/(k^2 - n^2). - _Antonio Graciá Llorente_, Sep 08 2024

%e G.f.: 1 + 2*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 + 924*x^6 + ...

%e For n=2, a(2) = 4!/(2!)^2 = 24/4 = 6, and this is the middle coefficient of the binomial expansion (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4. - _Michael B. Porter_, Jul 06 2016

%p A000984 := n-> binomial(2*n,n); seq(A000984(n), n=0..30);

%p with(combstruct); [seq(count([S,{S=Prod(Set(Z,card=i),Set(Z,card=i))}, labeled], size=(2*i)), i=0..20)];

%p with(combstruct); [seq(count([S,{S=Sequence(Union(Arch,Arch)), Arch=Prod(Epsilon, Sequence(Arch),Z)},unlabeled],size=i), i=0..25)];

%p with(combstruct):bin := {B=Union(Z,Prod(B,B))}: seq (count([B,bin,unlabeled],size=n)*n, n=1..25); # _Zerinvary Lajos_, Dec 05 2007

%p A000984List := proc(m) local A, P, n; A := [1,2]; P := [1];

%p for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), 2*P[-1]]);

%p A := [op(A), 2*P[-1]] od; A end: A000984List(28); # _Peter Luschny_, Mar 24 2022

%t Table[Binomial[2n, n], {n, 0, 24}] (* _Alonso del Arte_, Nov 10 2005 *)

%t CoefficientList[Series[1/Sqrt[1-4x],{x,0,25}],x]  (* _Harvey P. Dale_, Mar 14 2011 *)

%o (Magma) a:= func< n | Binomial(2*n,n) >; [ a(n) : n in [0..10]];

%o (PARI) A000984(n)=binomial(2*n,n) \\ much more efficient than (2n)!/n!^2. \\ _M. F. Hasler_, Feb 26 2014

%o (PARI) fv(n,p)=my(s);while(n\=p,s+=n);s

%o a(n)=prodeuler(p=2,2*n,p^(fv(2*n,p)-2*fv(n,p))) \\ _Charles R Greathouse IV_, Aug 21 2013

%o (PARI) fv(n,p)=my(s);while(n\=p,s+=n);s

%o a(n)=my(s=1);forprime(p=2,2*n,s*=p^(fv(2*n,p)-2*fv(n,p)));s \\ _Charles R Greathouse IV_, Aug 21 2013

%o (Haskell)

%o a000984 n = a007318_row (2*n) !! n  -- _Reinhard Zumkeller_, Nov 09 2011

%o (Maxima) A000984(n):=(2*n)!/(n!)^2$ makelist(A000984(n),n,0,30); /* _Martin Ettl_, Oct 22 2012 */

%o (Python)

%o from __future__ import division

%o A000984_list, b = [1], 1

%o for n in range(10**3):

%o     b = b*(4*n+2)//(n+1)

%o     A000984_list.append(b) # _Chai Wah Wu_, Mar 04 2016

%o (GAP) List([1..1000], n -> Binomial(2*n,n)); # _Muniru A Asiru_, Jan 30 2018

%Y Cf. A000108, A002420, A002457, A030662, A002144, A135091, A081696, A182400. Differs from A071976 at 10th term.

%Y Bisection of A001405 and of A226302. See also A025565, the same ordered partitions but without all in which are two successive zeros: 11110 (5), 11200 (18), 13000 (2), 40000 (0) and 22000 (1), total 26 and A025565(4)=26.

%Y Cf. A226078, A051924 (first differences).

%Y Row sums of A059481, A008459, A152229, A158815, A205946.

%Y Cf. A258290 (arithmetic derivative). Cf. A098616, A214377.

%Y See A261009 for a conjecture about this sequence.

%Y Cf. A046521 (first column).

%Y The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

%Y Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.

%Y Cf. A000346, A001700, A001791, A008549, A032443, A073016, A097805, A126869.

%Y Cf. A056040, A001316, A261130, A356637.

%K nonn,easy,core,nice,walk,frac

%O 0,2

%A _N. J. A. Sloane_