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A000984
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Central binomial coefficients: C(2*n,n) = (2*n)!/(n!)^2.
(Formerly M1645 N0643)
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468
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1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, 137846528820, 538257874440, 2104098963720, 8233430727600, 32247603683100
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refs;
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internal format)
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OFFSET
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0,2
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COMMENTS
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Devadoss refers to these numbers as type B Catalan numbers (cf. A000108).
Equal to the binomial coefficient sum Sum_{k=0..n} binomial(n,k)^2.
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
Convolving a(n) with itself yields A000302, the powers of 4. - T. D. Noe, Jun 11 2002
a(n)=Max{ (i+j)!/(i!j!) | 0<=i,j<=n } - Benoit Cloitre, May 30 2002
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
Also number of directed, convex polyominoes having semiperimeter n+2.
Also number of diagonally symmetric, directed, convex polyominoes having semiperimeter 2n+2. - Emeric Deutsch, Aug 03 2002
Also Sum_{k=0..n} binomial(n+k-1,k). - Vladeta Jovovic, Aug 28 2002
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
Number of possible values of a 2*n bit binary number for which half the bits are on and half are off. - Gavin Scott (gavin(AT)allegro.com), Aug 09 2003
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
Number of non-decreasing 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
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
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
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.
Number of standard tableaux of shape (n+1, 1^n). - Emeric Deutsch, May 13 2004
Erdos, Graham et al. conjectured that a(n) is never squarefree for sufficiently large n. Sarkozy 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. A000984(n)/(n+1) = A000108(n), that is, dividing by (n+1) scales the central binomial coefficients to Catalan numbers also called Segner numbers. - Jonathan Vos Post, Dec 04 2004
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
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
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
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
The number of walks of length 2n on an infinite linear lattice that begin and end at the origin. - Stefan Hollos (stefan(AT)exstrom.com), Dec 10 2007
The number of lattice paths from (0,0) to (n,n) using steps (1,0) and (0,1). [Joerg Arndt, Jul 01 2011]
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
Define the array m(1,j)=1 ; m(i,1)=1 ; m(i,j)=m(i,j-1) + m(j,i-1), then a(n) = m(n,n) [From philippe lallouet (philip.lallouet(AT)orange.fr), Sep 15 2008]
Also the Catalan transform of A000079. [R. J. Mathar, Nov 06 2008]
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. Finally, we remark that Erdos et al. conjectured that the central binomial coefficients C_n are never squarefree for n > 4 which has been proved by Granville and Ramare." [Jonathan Vos Post, Nov 14 2008]
Equals row sums of triangle A152229 [Gary W. Adamson, Nov 29 2008]
Equals row sums of triangle A158815 [Gary W. Adamson, Mar 27 2009]
This sequence appears in formulae in the link cited. [Oktay Haracci (timetunnel3(AT)hotmail.com), Apr 02 2009]
Equals INVERT transform of A081696: (1, 1, 3, 9, 29, 97, 333,...). [Gary W. Adamson, May 15 2009]
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]
Index the central binomial coefficients with the natural numbers 1,2,3...,n. It appears that dividing the central binomial coefficients by their indexes yields the Catalan numbers (A000108). [Jason Richardson-White (coyoteworks(AT)gmail.com), Jun 15 2009]
Number of nXn 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]
Hankel transform is 2^n. [Paul Barry, Aug 05 2009]
It appears that a(n) is also the number of quivers in the mutation class of twisted type BC_n for n>=2.
Central terms of Pascal's triangle: a(n) = A007318(2*n,n). [Reinhard Zumkeller, Nov 09 2011]
Equals row sums of triangle A205946. - Gary W. Adamson, Feb 01 2012
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]
Also a(n) = Product_{k=1..n} (4 - 2/k). [Michel Lagneau, Apr 28 2012]
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.
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
a(n) == 2 mod n^3 iff n is a prime > 3. (See Mestrovic link, p.4) - Gary Detlefs, Feb 16 2013
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]
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REFERENCES
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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.
M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
P. Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, Vol. 15 2012, #12.8.2. - From N. J. A. Sloane, Dec 29 2012
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.- From N. J. A. Sloane, Oct 08 2012
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 160.
A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining convex permutominoes, preprint, 2007.
Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, http://carma.newcastle.edu.au/~jb616/wmi-paper.pdf.
H. J. Brothers, Pascal's Prism: Supplementary Material, http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf.
Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3. Thierry Dana-Picard, Sequences of Definite Integrals, Factorials and Double Factorials, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.6.
D. Daly and L. Pudwell, Pattern avoidance in rook monoids, 2013; http://faculty.valpo.edu/lpudwell/slides/sandiego2013.pdf. - From N. J. A. Sloane, Feb 03 2013
E. Deutsch, Enumerating symmetric directed convex polyominoes, Discrete Math., 280 (2004), 225-231.
E. Deutsch and L. Shapiro, Seventeen Catalan identities, Bulletin of the Institute of Combinatorics and its Applications, 31, 31-38, 2001.
Devadoss, Satyan L., A realization of graph associahedra. Discrete Math. 309 (2009), no. 1, 271-276.
R. Duarte and A. G. de Oliveira, Short note on the convolution of binomial coefficients, arXiv preprint arXiv:1302.2100, 2013
Erdos, P.; Graham, R. L.; Ruzsa, I. Z.; and Straus, E. G. "On the Prime Factors of C(2n,n)." Math. Comput. 29, 83-92, 1975.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, Arxiv preprint arXiv:1203.6792, 2012. - From N. J. A. Sloane, Oct 03 2012
Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, Arxiv preprint arXiv:1203.1476, 2012. - From N. J. A. Sloane, Sep 14 2012
H. W. Gould, Combinatorial Identities, Morgantown, 1972, (3.66), page 30.
M. Griffiths, The Backbone of Pascal's Triangle, United Kingdom Mathematics Trust (2008), 3-124. [From Martin Griffiths (griffm(AT)essex.ac.uk), Mar 28 2009]
Granville, A. and Ramare, O. "Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients." Mathematika 43, 73-107, 1996.
T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150, 2013
D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012. - From N. J. A. Sloane, Sep 16 2012
A. Petojevic and N. Dapic, The vAm(a,b,c;z) function, Preprint 2013, http://www.mi.sanu.ac.rs/~gvm/radovi/AP-Budva.pdf
Sarkozy, A. "On Divisors of Binomial Coefficients. I." J. Number Th. 20, 70-80, 1985.
L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012. - From N. J. A. Sloane, Dec 16 2012
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
D. H. Bailey, J. M. Borwein and D. M. Bradley, Experimental determination of Ap'ery-like identities for zeta(4n+2)
Robert J. Betts, Lack of Divisibility of {2N choose N} by three fixed odd primes infinitely often, through the Extension of a Result by P. Erdos, et al, Oct 15, 2010. [From Jonathan Vos Post, Oct 17 2010]
J. Borwein and D. Bradley, Empirically determined Ap'ery-like formulae for zeta(4n+3)
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.
N. T. Cameron, Random walks, trees and extensions of Riordan group techniques
J. Cigler, Some nice Hankel determinants, arXiv:1109.1449, 2011.
B. N. Cooperstein and E. E. Shult, A note on embedding and generating dual polar spaces. Adv. Geom. 1 (2001), 37-48. See Theorem 5.4.
R. M. Dickau, Shortest-path diagrams
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv:1110.6638, 2011
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 77
Oktay Haracci (timetunnel3(AT)hotmail.com), Regular Polygons
R. H. Hardin, Binary arrays with both rows and cols sorted, symmetries
Milan Janjic, Two Enumerative Functions
I. Jensen, Series exapansions for self-avoiding polygons
C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, Arxiv preprint arXiv:1201.1323, 2012
V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv:1206.0877v1
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), Arxiv preprint arXiv:1111.3057, 2011
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Armin Straub, Tewodros Amdeberhan and Victor H. Moll, The p-adic valuation of k-central binomial coefficients, Nov 13, 2008, pp. 10-11. [From Jonathan Vos Post, Nov 14 2008]
V. Strehl, Recurrences and Legendre transform
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.
H. A. Verrill, Sums of squares of binomial coefficients, ...
Eric Weisstein's World of Mathematics, Binomial Sums
Eric Weisstein's World of Mathematics, Central Binomial Coefficient
Eric Weisstein's World of Mathematics, Staircase Walk
Eric Weisstein's World of Mathematics, Circle Line Picking
Index entries for "core" sequences
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FORMULA
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G.f.: A(x) = (1 - 4*x)^(-1/2) = 1 + 2*x + 6*x^2 + 20*x^3 + ...
a(n) = 2^n/n! * product(k=0..n-1, (2*k+1) ).
a(n) = a(n-1)*(4-2/n) = 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
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
Integral representation as n-th moment of a positive function on the interval[0, 4], in Maple notation: a(n)= int(x^n*((x*(4-x))^(-1/2))/Pi, x=0..4), n=0, 1, ... This representation is unique. - Karol A. Penson, Sep 17 2001
sum(n>=1, 1/a(n))=(2*Pi*sqrt(3)+9)/27 [Lehmer, Am. Math. Monthly 92 (1985) 449, eq (15)] - Benoit Cloitre, May 01 2002
E.g.f.: exp(2*x)*I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 08 2002
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.
a(n) = sum(k=0, n, C(n, k)^2). - Benoit Cloitre, Jan 31 2003
Determinant of n X n matrix M(i, j)=binomial(n+i, j) - Benoit Cloitre, Aug 28 2003
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
a(n) = 2*Sum{k= 0...(n-1), a(k)*a(n-k+1)/(k+1)}. - Philippe Deléham, Jan 01 2004
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
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
a(n)=sum{k=0..n, binomial(2n+1, k)*sin((2n-2k+1)*pi/2)}. - Paul Barry, Nov 02 2004
a(n-1)=(1/2)*(-1)^n*sum_{0<=i, j<=n}(-1)^(i+j)*binomial(2n, i+j) - Benoit Cloitre, Jun 18 2005
a(n) = C(2n, n-1) + C(n) = A001791(n) + A000108(n). - Lekraj Beedassy, Aug 02 2005
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
a(n)=A006480(n)/A005809(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007
a(n)=Sum{k, 0<=k<=n}A106566(n,k)*2^k. - Philippe DELEHAM, Aug 25 2007
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
a(n)=4^n*sum{k=0..n, C(n,k)(-4)^(-k)*A000108(n+k)}; - Paul Barry, Oct 18 2007
Row sums of triangle A135091 - Gary W. Adamson, Nov 18 2007
a(n)=Sum_{k, 0<=k<=n}A039598(n,k)*A059841(k). [Philippe DELEHAM, Nov 12 2008]
A007814(a(n))=A000120(n). [Vladimir Shevelev, Jul 20 2009]
From Paul Barry, Aug 05 2009: (Start)
G.f.: 1/(1-2x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction);
G.f.: 1/(1-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)
a(n)=Product(k=1..n)[4-2/k] [David Brown, Sep 19 2009]
If n>=3 is prime, then a(n)==2(mod 2*n). [Vladimir Shevelev, Sep 05 2010]
Let A(x) be the g.f. and B(x)=A(-x), then B(x)=sqrt(1-4*x*B(x)^2) [From Vladimir Kruchinin, Jan 16 2011]
a(n)=(-4)^n*sqrt(Pi)/(gamma((1/2-n))*gamma(1+n)) [Gerry Martens, May 03 2011]
a(n) = upper left term in M^n, M = the infinite square production matrix:
2, 2, 0, 0, 0, 0,...
1, 1, 1, 0, 0, 0,...
1, 1, 1, 1, 0, 0,...
1, 1, 1, 1, 1, 0,...
1, 1, 1, 1, 1, 1,....
- Gary W. Adamson, Jul 14 2011
a(n) = Hypergeometric([-n,-n],[1],1). - Peter Luschny, Nov 01 2011
E.g.f.: exp(2*x)*I_0(2*x)=1+2*x/(Q(0)-2*x); Q(k)=2*x*(2*k+1)+(k+1)^2-2*x*(2*k+3)*(k+1)^2/Q(k+1), where I_0 is Bessel function; (continued fraction). - Sergei N. Gladkovskii, Nov 22 2011
E.g.f.: hypergeometric([1/2],[1],4*x), simplified the e.g.f. given above, also by M. Somos. - Wolfdieter Lang, Jan 13 2012.
a(n) = 2*Sum_{k=0..n-1} (a(k)*A000108(n-k-1)). - Alzhekeyev Ascar M., Mar 09 2012
G.f.: 1/sqrt(1-4*x)= 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
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
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
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
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MAPLE
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A000984 := n-> binomial(2*n, n);
with(combstruct); [seq(count([S, {S=Prod(Set(Z, card=i), Set(Z, card=i))}, labeled], size=(2*i)), i =0..20)];
with(combstruct); [seq(count([S, {S=Sequence(Union(Arch, Arch)), Arch=Prod(Epsilon, Sequence(Arch), Z)}, unlabeled], size=i), i=0..25)];
Z:=(1-sqrt(1-z))*4^n/sqrt(1-z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=0..24); # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 01 2007
with(combstruct):bin := {B=Union(Z, Prod(B, B))}: seq (count([B, bin, unlabeled], size=n)*n, n=1..25); # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 05 2007
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MATHEMATICA
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Table[Binomial[2n, n], {n, 0, 24}] (* Alonso del Arte, Nov 10 2005 *)
CoefficientList[Series[1/Sqrt[1-4x], {x, 0, 25}], x] (* Harvey P. Dale, Mar 14 2011 *)
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PROG
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(MAGMA) a:= func< n | Binomial(2*n, n) >; [ a(n) : n in [0..10]];
(PARI) Vec(+O(x^99)) \\ Charles R Greathouse IV, Oct 23 2012
(PARI) A000984(n)=if(n<0, 0, (2*n)!/n!^2)
(Haskell)
a000984 n = a007318_row (2*n) !! n -- Reinhard Zumkeller, Nov 09 2011
(Maxima) A000984(n):=(2*n)!/(n!)^2$ makelist(A000984(n), n, 0, 30); [Martin Ettl, Oct 22 2012]
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CROSSREFS
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a(n+1)=2*A001700(n)=A030662(n)+1. a(2*n) = A001448(n), a(2*n+1) = 2*A002458(n).
Cf. A000108, A002420, A002457, A030662, A002144, A135091, A152229, A158815, A081696, A205946, A182400. Differs from A071976 at 10-th term.
Bisection of A001405. Row sums of A059481 and of A008459. 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.
Sequence in context: A056616 A065346 A071976 * A087433 A119373 A151284
Adjacent sequences: A000981 A000982 A000983 * A000985 A000986 A000987
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KEYWORD
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nonn,easy,core,nice,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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