

A001700


a(n) = binomial(2*n+1, n+1): number of ways to put n+1 indistinguishable balls into n+1 distinguishable boxes = number of (n+1)st degree monomials in n+1 variables = number of monotone maps from 1..n+1 to 1..n+1.
(Formerly M2848 N1144)


355



1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052
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OFFSET

0,2


COMMENTS

To show for example that C(2n+1, n+1) is the number of monotone maps from 1..n + 1 to 1..n + 1, notice that we can describe such a map by a nondecreasing sequence of length n + 1 with entries from 1 to n + 1. The number k of increases in this sequence is anywhere from 0 to n. We can specify these increases by throwing k balls into n+1 boxes, so the total is Sum_{k = 0..n} C((n+1) + k  1, k) = C(2*n+1, n+1).
Also number of ordered partitions (or compositions) of n + 1 into n + 1 parts. E.g., a(2) = 10: 003, 030, 300, 012, 021, 102, 120, 210, 201, 111.  Mambetov Bektur (bektur1987(AT)mail.ru), Apr 17 2003
Also number of walks of length n on square lattice, starting at origin, staying in first and second quadrants.  David W. Wilson, May 05 2001. (E.g., for n = 2 there are 10 walks, all starting at 0, 0: 0, 1 > 0, 0; 0, 1 > 1, 1; 0, 1 > 0, 2; 1, 0 > 0, 0; 1, 0 > 1, 1; 1, 0 > 2, 0; 1, 0 > 1, 1; 1, 0 > 0, 0; 1, 0 > 1, 1; 1, 0> 2, 0.)
Also total number of leaves in all ordered trees with n + 1 edges.
Also number of digitally balanced numbers [A031443] from 2^(2*n+1) to 2^(2*n+2).  Naohiro Nomoto, Apr 07 2001
Also number of ordered trees with 2*n + 2 edges having root of even degree and nonroot nodes of outdegree 0 or 2.  Emeric Deutsch, Aug 02 2002
Also number of paths of length 2*d(G) connecting two neighboring nodes in optimal chordal graph of degree 4, G(2*d(G)^2 + 2*d(G) + 1, 2d(G) + 1), where d(G) = diameter of graph G.  S. Bujnowski (slawb(AT)atr.bydgoszcz.pl), Feb 11 2002
Define an array by m(1, j) = 1, m(i, 1) = i, m(i, j) = m(i, j1) + m(i1, j); then a(n) = m(n, n), diagonal of A165257  Benoit Cloitre, May 07 2002
Also the numerator of the constant term in the expansion of cos^(2*n)(x) or sin^(2*n)(x) when the denominator is 2^(2*n1).  Robert G. Wilson v
Consider the expansion of cos^n(x) as a linear combination of cosines of multiple angles. If n is odd, then the expansion is a combination of a*cos((2*k1)*x)/2^(n1) for all 2*k  1 <= n. If n is even, then the expansion is a combination of a*cos(2k*x)/2^(n1) terms plus a constant. "The constant term, [a(n)/2^(2n1)], is due to the fact that [cos^2n(x)] is never negative, i.e., electrical engineers would say the average or 'dc value' of [cos^(2*n)(x)] is [a(n)/2^(2*n1)]. The dc value of [cos^(2*n1)(x)] on the other hand, is zero because it is symmetrical about the horizontal axis, i.e., it is negative and positive equally." Nahin[62]  Robert G. Wilson v, Aug 01 2002
Also number of times a fixed Dyck word of length 2*k occurs in all Dyck words of length 2*n + 2*k. Example: if the fixed Dyck word is xyxy (k = 2), then it occurs a(1) = 3 times in the 5 Dyck words of length 6 (n = 1): (xy[xy)xy], xyxxyy, xxyyxy, x(xyxy)y, xxxyyy (placed between parentheses).  Emeric Deutsch, Jan 02 2003
a(n+1) is the determinant of the n X n matrix m(i, j) = binomial(2*ni, j).  Benoit Cloitre, Aug 26 2003
a(n1) = (2*n)!/(2*n!*n!), formula in [Davenport] used by Gauss for the special case prime p = 4*n + 1: x = a(n1) mod p and y = x*(2n)! mod p are solutions of p = x^2 + y^2.  Frank Ellermann. Example: For prime 29 = 4*7 + 1 use a(71) = 1716 = (2*7)!/(2*7!*7!), 5 = 1716 mod 29 and 2 = 5*(2*7)! mod 29, then 29 = 5*5 + 2*2.
The number of compositions of 2*n, say c_1 + c_2 + ... + c_k = 2n, satisfy that Sum_{i = 1..j} c_i < 2*j for all j = 1..k, or equivalently, the number of subsets, say S, of [2*n1] = {1, 2, ..., 2*n1} with at least n elements such that if 2k is in S, then there must be at least k elements in S smaller than 2k. E.g., a(2) = 3 because we can write 4 = 1 + 1 + 1 + 1 = 1 + 1 + 2 = 1 + 2 + 1.  Ricky X. F. Chen (ricky_chen(AT)mail.nankai.edu.cn), Jul 30 2006
The number of walks of length 2*n + 1 on an infinite linear lattice that begin at the origin and end at node (1). Also the number of paths on a square lattice from the origin to (n+1, n) that use steps (1,0) and (0,1). Also number of binary numbers of length 2*n + 1 with n + 1 ones and n zeros.  Stefan Hollos (stefan(AT)exstrom.com), Dec 10 2007
If Y is a 3subset of an 2*nset X then, for n >= 3, a(n1) is the number of nsubsets of X having at least two elements in common with Y.  Milan Janjic, Dec 16 2007
Also the number of rankings (preferential arrangements) of n unlabeled elements onto n levels when empty levels are allowed.  Thomas Wieder, May 24 2008
Also the Catalan transform of A000225 shifted one index, i.e., dropping A000225(0).  R. J. Mathar, Nov 11 2008
With offset 1. The number of solutions in nonnegative integers to X1 + X2 + ... + Xn = n. The number of terms in the expansion of (X1 + X2 + ... + Xn)^n. The coefficient of x^n in the expansion of (1 + x + x^2 + ...)^n. The number of distinct image sets of all functions taking [n] into [n].  Geoffrey Critzer, Feb 22 2009
The Hankel transform of the aerated sequence 1, 0, 3, 0, 10, 0, ... is 1, 3, 3, 5, 5, 7, 7, ... (A109613(n+1)).  Paul Barry, Apr 21 2009
Also the number of distinct network topologies for a network of n items with 1 to n  1 unidirectional connections to other objects in the network.  Anthony Bachler, May 05 2010
Equals INVERT transform of the Catalan numbers starting with offset 1. E.g.: a(3) = 35 = (1, 2, 5) dot (10, 3, 1) + 14 = 21 + 14 = 35.  Gary W. Adamson, May 15 2009
The integral of 1/(1+x^2)^(n+1) is given by a(n)/2^(2*n  1) * (x/(1 + x^2)^n*P(x) + arctan(x)), where P(x) is a monic polynomial of degree 2*n  2 with rational coefficients.  Christiaan van de Woestijne, Jan 25 2011
a(n) is the number of Schroder paths of semilength n in which the (2,0)steps at level 0 come in 2 colors and there are no (2,0)steps at a higher level. Example: a(2) = 10 because, denoting U = (1,1), H = (1,0), and D = (1,1), we have 2^2 = 4 paths of shape HH, 2 paths of shape HUD, 2 paths of shape UDH, and 1 path of each of the shapes UDUD and UUDD.  Emeric Deutsch, May 02 2011
a(n) is the number of Motzkin paths of length n in which the (1,0)steps at level 0 come in 3 colors and those at a higher level come in 2 colors. Example: a(3)=35 because, denoting U = (1,1), H = (1,0), and D = (1,1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, 3 paths of shape UDH, and 2 paths of shape UHD.  Emeric Deutsch, May 02 2011
Also number of digitally balanced numbers having length 2*(n + 1) in binary representation: a(n) = #{m: A070939(A031443(m)) = 2*(n + 1)}.  Reinhard Zumkeller, Jun 08 2011
a(n) equals 2^(2*n + 3) times the coefficient of Pi in 2F1([1/2, n+2]; [3/2]; 1).  John M. Campbell, Jul 17 2011
For positive n, a(n) equals 4^(n+2) times the coefficient of Pi^2 in Integral_{x = 0..Pi/2} x sin^(2*n + 2)x.  John M. Campbell, Jul 19 2011 [Apparently, the contributor means Integral_{x = 0..Pi/2} x * (sin(x))^(2*n + 2).]
a(n1) = C(2*n, n)/2 is the number of ways to assign 2*n people into 2 (unlabeled) groups of size n.  Dennis P. Walsh, Nov 09 2011
Equals row sums of triangle A205945.  Gary W. Adamson, Feb 01 2012
a(n1) gives the number of nregular sequences defined by Erdős and Gallai in 1960 in connection with the degree sequences of simple graphs.  Matuszka Tamás, Mar 06 2013
a(n) is the sum of falling diagonals of squares in the comment in A085812 (equivalent to the Cloitre formula of Aug 2002).  John Molokach, Sep 26 2013
For n > 0: largest terms of Zigzag matrices as defined in A088961.  Reinhard Zumkeller, Oct 25 2013
Also the number of different possible win/loss round sequences (from the perspective of the eventual winner) in a "best of 2*n + 1" twoplayer game. For example, a(2) = 10 means there are 10 different win/loss sequences in a "best of 5" game (like a tennis match in which the first player to win 3 sets, out of a maximum of 5, wins the match); the 10 sequences are WWW, WWLW, WWLLW, WLWW, WLWLW, WLLWW, LWWW, LWWLW, LWLWW, LLWWW. See also A072600.  Philippe Beaudoin, May 14 2014; corrected by Jon E. Schoenfield, Nov 23 2014
When adding 1 to the beginning of the sequence: Convolving a(n)/2^n with itself equals 2^(n+1). For example, when n = 4: convolving {1, 1/1, 3/2, 10/4, 35/8, 126/16} with itself is 32 = 2^5.  Bob Selcoe, Jul 16 2014
From Tom Copeland, Nov 09 2014: (Start)
The shifted array belongs to a family of arrays associated to the Catalan A000108 (t = 1), and Riordan, or Motzkin sums A005043 (t = 0), with the o.g.f. [1  sqrt(1  4x/(1 + (1  t)x))]/2 and inverse x*(1  x)/[1 + (t  1)*x*(1  x)]. See A091867 for more info on this family. Here is t = 3 (mod signs in the results).
Let C(x) = [1  sqrt(14x)]/2, an o.g.f. for the Catalan numbers A000108, with inverse Cinv(x) = x*(1x) and P(x,t) = x/(1 + t*x) with inverse P(x, t).
O.g.f: G(x) = [1 + sqrt(1 + 4*x/(1  4*x))]/2 = C[P(x, 4)].
Inverse o.g.f: Ginv(x) = x*(1 + x)/(1 + 4*x*(1 + x)) = P(Cinv(x), 4) (shifted signed A001792). A088218(x) = 1 + G(x).
Equals A001813/2 omitting the leading 1 there. (End)
Placing n distinguishable balls into n indistinguishable boxes gives A000110(n) (the number of set partitions).  N. J. A. Sloane, Jun 19 2015
The sequence is the INVERTi transform of A049027: (1, 4, 17, 74, 326, ...).  Gary W. Adamson, Jun 23 2015
a(n) is the number of compositions of 2*n + 2 such that the sum of the elements at odd positions is equal to the sum of the elements at even positions. a(2) = 10 because there are 10 such compositions of 6: (3, 3), (1, 3, 2), (2, 3, 1), (1, 1, 2, 2), (1, 2, 2, 1), (2, 2, 1, 1), (2, 1, 1, 2), (1, 2, 1, 1, 1), (1, 1, 1, 2, 1), (1, 1, 1, 1, 1, 1).  Ran Pan, Oct 08 2015
a(n1) is also the Schur function of the partition (n) of n evaluated at x_1 = x_2 = ... = x_n = 1, i.e., the number of semistandard Young tableaux of shape (n) (weakly increasing rows with n boxes with numbers from {1, 2, ..., n}).  Wolfdieter Lang, Oct 11 2015
Also the number of ordered (rooted planar) forests with a total of n+1 edges and no trivial trees.  Nachum Dershowitz, Mar 30 2016
a(n) is the number of sets (i1,...in) of length n so that n >= i1 >= i2 >= ...>= in >= 1. For instance, n=3 as there are only 10 such sets (3,3,3) (3,3,2) (3,3,1) (3,2,2) (3,2,1) (3,1,1) (2,2,2) (2,2,1) (2,1,1) (1,1,1,) 3,2,1 is each used 10 times respectively.  Anton Zakharov, Jul 04 2016
The repeated middle term in the odd rows of Pascal's triangle, or half the central binomial coefficient in the even rows of Pascal's triangle, n >= 2.  Enrique Navarrete, Feb 12 2018
a(n) is the number of walks of length 2n+1 from the origin with steps (1,1) and (1,1) that stay on or above the xaxis. Equivalently, a(n) is the number of walks of length 2n+1 from the origin with steps (1,0) and (0,1) that stay in the first octant.  Alexander Burstein, Dec 24 2019
Total number of nodes summed over all Dyck paths of semilength n.  Alois P. Heinz, Mar 08 2020
a(n1) is the determinant of the n X n matrix m(i, j) = binomial(n+i1, j).  Fabio Visonà, May 21 2022


REFERENCES

H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, ch. V.3 (p. 122).
A. Frosini, R. Pinzani and S. Rinaldi, About half the middle binomial coefficient, Pure Math. Appl., 11 (2000), 497508.
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.
J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
Paul J. Nahin, "An Imaginary Tale, The Story of [Sqrt(1)]," Princeton University Press, Princeton, NJ 1998, p. 62.
L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 3346.
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).


LINKS

Matuszka Tamás, Table of n, a(n) for n = 0..1200 (first 100 terms from T. D. Noe)
J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011), #11.2.6, theorem 4.
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On OneParameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
Martin Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 25442563.
Elena Barcucci, Andrea Frosini and Simone Rinaldi, On directedconvex polyominoes in a rectangle, Discr. Math., 298 (2005), 6278.
Elena Barcucci, Antonio Bernini, and Renzo Pinzani, Exhaustive generation of positive lattice paths, Semantic Sensor Networks Workshop 2018, CEUR Workshop Proceedings (2018), Vol. 2113.
JeanLuc Baril, David Bevan, and Sergey Kirgizov, Bijections between directed animals, multisets and GrandDyck paths, arXiv:1906.11870 [math.CO], 2019.
Paul Barry, On IntegerSequenceBased Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, 9 (2006), Article 06.2.4.
Paul Barry, The Central Coefficients of a Family of Pascallike Triangles and Colored Lattice Paths, J. Int. Seq., 22 (2019), Article 19.1.3.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Antonio Bernini, Filippo Disanto, Renzo Pinzani and Simone Rinaldi, Permutations defining convex permutominoes, J. Int. Seq. 10 (2007), #07.9.7.
Ciprian Borcea and Ileana Streinu, On the number of embeddings of minimally rigid graphs, arXiv:math/0207126 [math.MG], 2002.
Mireille BousquetMélou, New enumerative results on twodimensional directed animals, Discr. Math., 180 (1998), 73106. See Eq. (1).
M. BousquetMélou and A. Rechnitzer, Lattice animals and heaps of dimers.
JeanPaul Bultel and Samuele Giraudo, Combinatorial Hopf algebras from PROs, arXiv:1406.6903 [math.CO], 2014. [DOI]
Libor Caha and Daniel Nagaj, The pairflip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quantph], 2018.
David Callan, A Combinatorial Interpretation for a SuperCatalan Recurrence, Journal of Integer Sequences, 8 (2005), Article 05.1.8.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs., 3 (2000), #00.1.5.
GiSang Cheon, Hana Kim, and Louis W. Shapiro, Mutation effects in ordered trees, arXiv:1410.1249 [math.CO], 2014.
Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, and Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, 18 (2015), Article 15.5.8.
Nachum Dershowitz, 1700 Forests, arXiv:1608.08740 [cs.DM], 2016.
Leonard E. Dickson, On the Inscription of Regular Polygons, Annals of Mathematics, Vol. 9, No. 1/6 (1894  1895), pp. 7384.
Leonard E. Dickson, Problem 44, The American Mathematical Monthly, Vol. 2, No. 7/8 (Jul.  Aug., 1895), pp. 229230.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Further Results on Paths in an nDimensional Cubic Lattice, Journal of Integer Sequences, 21 (2018), Article 18.1.2.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
N. Gromov and P. Vieira, Tailoring ThreePoint Functions and Integrability IV. Thetamorphism, arXiv:1205.5288 [hepth], 2012.  From N. J. A. Sloane, Oct 23 2012
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, 3 (2000), Article #00.1.6.
GuoNiu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
GuoNiu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 145.
A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4(2) (2012) 260288.
Milan Janjic, Two Enumerative Functions.
Christian Krattenthaler and Daniel Yaqubi, Some determinants of path generating functions, II, Adv. Appl. Math. 101 (2018), 232265.
Dmitry Kruchinin, Superposition's properties of logarithmic generating functions, arXiv:1109.1683 [math.CO], 20112015.
Markus Kuba and Alois Panholzer, Stirling permutations containing a single pattern of length three, Australasian Journal of Combinatorics 74(2) (2019), 216239.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., 3 (2000), #00.2.4.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
H. Li and T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, example 40.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
T. Mansour and M. Shattuck, A statistic on ncolor compositions and related sequences, Proc. Indian Acad. Sci. (Math. Sci.) 124(2) (2014), pp. 127140.
M. D. McIlroy, Letter to N. J. A. Sloane (no date).
Donatella Merlini and Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, 21 (2018), Article 18.1.3.
Hanna Mularczyk, Lattice Paths and PatternAvoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
A. Nkwanta, Riordan matrices and higherdimensional lattice walks, J. Stat. Plann. Infer. 140 (2010), 23212334.
Asamoah Nkwanta and Earl R. Barnes, Two Catalantype Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, 15 (2012), Article 12.3.3.  From N. J. A. Sloane, Sep 16 2012
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., 4 (2001), #01.2.1.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their Nnumbers, not their Anumbers.
Louis Shapiro, Problem 10753 Amer. Math. Monthly, 106(8) (1999), p. 777.
Louis Shapiro et al., Leaves of Ordered Trees: 10753, Amer. Math. Monthly, 108(9) (Nov., 2001), 873874.
Eric Weisstein's World of Mathematics, Binomial Coefficient.
Eric Weisstein's World of Mathematics, Odd Graph.
Index entries for "core" sequences


FORMULA

a(n1) = binomial(2*n, n)/2 = (2*n)!/(2*n!*n!).
Dfinite with recurrence: a(0) = 1, a(n) = 2*(2*n+1)*a(n1)/(n+1) for n > 0.
G.f.: (1/sqrt(1  4*x)  1)/(2*x).
L.g.f. log((1  sqrt(1  4*x))/(2*x)) = Sum_{n > 0} a(n)/n*x^n.  Vladimir Kruchinin, Aug 10 2010
G.f.: 2F1([1, 3/2]; [2]; 4*x).  Paul Barry, Jan 23 2009
G.f.: 1/(1  2*x  x/(1  x/(1  x/(1  x/(1  ... (continued fraction).  Paul Barry, May 06 2009
G.f.: c(x)^2/(1  x*c(x)^2), c(x) the g.f. of A000108.  Paul Barry, Sep 07 2009
O.g.f.: c(x)/sqrt(1  4*x) = (2  c(x))/(1  4*x), with c(x) the o.g.f. of A000108. Added second formula.  Wolfdieter Lang, Sep 02 2012
Convolution of A000108 (Catalan) and A000984 (central binomial): Sum_{k=0..n} C(k)*binomial(2*(nk), nk), C(k) Catalan.  Wolfdieter Lang, Dec 11 1999
a(n) = Sum_{k=0..n} C(n+k, k).  Benoit Cloitre, Aug 20 2002
a(n) = Sum_{k=0..n} C(n, k)*C(n+1, k+1).  Benoit Cloitre, Oct 19 2002
a(n) = Sum_{k = 0..n+1} binomial(2*n+2, k)*cos((n  k + 1)*Pi).  Paul Barry, Nov 02 2004
a(n) = 4^n*binomial(n+1/2, n)/(n+1).  Paul Barry, May 10 2005
E.g.f.: Sum_{n >= 0} a(n)*x^(2*n + 1)/(2*n + 1)! = BesselI(1, 2*x).  Michael Somos, Jun 22 2005
E.g.f. in Maple notation: exp(2*x)*(BesselI(0, 2*x) + BesselI(1, 2*x)). Integral representation as nth moment of a positive function on [0, 4]: a(n) = int(x^n*((x/(4x))^(1/2)), x=0..4)/(2*Pi), n >= 0. This representation is unique.  Karol A. Penson, Oct 11 2001
Narayana transform of [1, 2, 3, ...]. Let M = the Narayana triangle of A001263 as an infinite lower triangular matrix and V = the Vector [1, 2, 3, ...]. Then A001700 = M * V.  Gary W. Adamson, Apr 25 2006
a(n) = A122366(n,n).  Reinhard Zumkeller, Aug 30 2006
a(n) = C(2*n, n) + C(2*n, n1) = A000984(n) + A001791(n).  Zerinvary Lajos, Jan 23 2007
a(n1) = (n+1)*(n+2)*...*(2*n1)/(n1)! (product of n1 consecutive integers, divided by (n1)!).  Jonathan Vos Post, Apr 09 2007; [Corrected and shortened by Giovanni Ciriani, Mar 26 2019]
a(n1) = (2*n  1)!/(n!*(n  1)!).  William A. Tedeschi, Feb 27 2008
a(n) = (2*n + 1)*A000108(n).  Paul Barry, Aug 21 2007
Binomial transform of A005773 starting (1, 2, 5, 13, 35, 96, ...) and double binomial transform of A001405.  Gary W. Adamson, Sep 01 2007
Row sums of triangle A132813.  Gary W. Adamson, Sep 01 2007
Row sums of triangle A134285.  Gary W. Adamson, Nov 19 2007
a(n) = 2*A000984(n)  A000108(n), that is, a(n) = 2*C(2*n, n)  nth Catalan number.  Joseph Abate, Jun 11 2010
Conjectured: 4^n GaussHypergeometric(1/2,n; 2; 1)  Solution for the path which stays in the first and second quadrant.  Benjamin Phillabaum, Feb 20 2011
a(n)= Sum_{k=0..n} A038231(n,k) * (1)^k * A000108(k).  Philippe Deléham, Nov 27 2009
Let A be the Toeplitz matrix of order n defined by: A[i,i1] = 1, A[i,j] = Catalan(ji), (i <= j), and A[i,j] = 0, otherwise. Then, for n >= 1, a(n) = (1)^n * charpoly(A,2).  Milan Janjic, Jul 08 2010
a(n) is the upper left term of M^(n+1), where M is the infinite matrix in which a column of (1,2,3,...) is prepended to an infinite lower triangular matrix of all 1's and the rest zeros, as follows:
1, 1, 0, 0, 0, ...
2, 1, 1, 0, 0, ...
3, 1, 1, 1, 0, ...
4, 1, 1, 1, 1, ...
...
Alternatively, a(n) is the upper left term of M^n where M is the infinite matrix:
3, 1, 0, 0, 0, ...
1, 1, 1, 0, 0, ...
1, 1, 1, 1, 0, ...
1, 1, 1, 1, 1, ...
...
 Gary W. Adamson, Jul 14 2011
a(n) = (n + 1)*hypergeom([n, n], [2], 1).  Peter Luschny, Oct 24 2011
a(n) = Pochhammer(n+1, n+1)/(n+1)!.  Peter Luschny, Nov 07 2011
E.g.f.: 1 + 6*x/(U(0)  6*x); U(k) = k^2 + (4*x + 3)*k + 6*x + 2  2*x*(k + 1)*(k + 2)*(2*k + 5)/U(k+1); (continued fraction).  Sergei N. Gladkovskii, Nov 18 2011
a(n) = 2*A000984(n)  A000108(n). [Abate & Whitt]
a(n) = 2^(2*n+1)*binomial(n+1/2, 1/2).  Peter Luschny, May 06 2014
For n > 1: a(n1) = A166454(2*n, n), central terms in A166454.  Reinhard Zumkeller, Mar 04 2015
a(n) = 2*4^n*Gamma(3/2 + n)/(sqrt(Pi)*Gamma(2+n)).  Peter Luschny, Dec 14 2015
a(n) ~ 2*4^n*(1  (5/8)/n + (73/128)/n^2  (575/1024)/n^3 + (18459/32768)/n^4)/sqrt(n*Pi).  Peter Luschny, Dec 16 2015
a(n) = (1)^(n)*B(n, n+1, n1)/n!, where B(n,a,x) is a generalized Bernoulli polynomial.  Vladimir Kruchinin, Apr 06 2016
a(n) = Gamma(2 + 2*n)/(n!*Gamma(2 + n)). Andres Cicuttin, Apr 06 2016
a(n) = (n + (n + 1))!/(Gamma(n)*Gamma(1 + n)*A002378(n)), for n > 0. Andres Cicuttin, Apr 07 2016
From Ilya Gutkovskiy, Jul 04 2016: (Start)
Sum_{n >= 0} 1/a(n) = 2*(9 + 2*sqrt(3)*Pi)/27 = A248179.
Sum_{n >= 0} (1)^n/a(n) = 2*(5 + 4*sqrt(5)*arcsinh(1/2))/25 = 2*(5*A145433  1).
Sum_{n >= 0} (1)^n*a(n)/n! = BesselI(2,2)*exp(2) = A229020*A092553. (End)
a(n) = Sum_{k=2^n..2^(n+1)1} A178244(k).  Mikhail Kurkov, Feb 20 2021


EXAMPLE

There are a(2)=10 ways to put 3 indistinguishable balls into 3 distinguishable boxes, namely, (OOO)()(), ()(OOO)(), ()()(OOO), (OO)(O)(), (OO)()(O), (O)(OO)(), ()(OO)(O), (O)()(OO), ()(O)(OO), and (O)(O)(O).  Dennis P. Walsh, Apr 11 2012
a(2) = 10: Semistandard Young tableaux for partition (3) of 3 (the indeterminates x_i, i = 1, 2, 3 are omitted and only their indices are given): 111, 112, 113, 122, 123, 133, 222, 223, 233, 333.  Wolfdieter Lang, Oct 11 2015


MAPLE

A001700 := n > binomial(2*n+1, n+1); seq(A001700(n), n=0..20);
A001700List := proc(m) local A, P, n; A := [1]; P := [1];
for n from 1 to m  2 do P := ListTools:PartialSums([op(P), 2*P[1]]);
A := [op(A), P[1]] od; A end: A001700List(27); # Peter Luschny, Mar 24 2022


MATHEMATICA

Table[ Binomial[2n + 1, n + 1], {n, 0, 23}]
CoefficientList[ Series[2/((Sqrt[1  4 x] + 1)*Sqrt[1  4 x]), {x, 0, 22}], x] (* Robert G. Wilson v, Aug 08 2011 *)


PROG

(Sage) [rising_factorial(n+1, n+1)/factorial(n+1) for n in (0..22)] # Peter Luschny, Nov 07 2011
(PARI) a(n)=binomial(2*n+1, n+1)
(Haskell)
a001700 n = a007318 (2*n+1) (n+1)  Reinhard Zumkeller, Oct 25 2013
(Magma) [Binomial(2*n, n)/2: n in [1..40]]; // Vincenzo Librandi, Nov 10 2014
(PARI) z='z+O('z^50); Vec((1/sqrt(14*z)1)/(2*z)) \\ Altug Alkan, Oct 11 2015
(Python)
from __future__ import division
A001700_list, b = [], 1
for n in range(10**3):
A001700_list.append(b)
b = b*(4*n+6)//(n+2) # Chai Wah Wu, Jan 26 2016
(Maxima)
B(n, a, x):=coeff(taylor(exp(x*t)*(t/(exp(t)1))^a, t, 0, 20), t, n)*n!;
makelist((1)^(n)*B(n, n+1, n1)/n!, n, 0, 10); /* Vladimir Kruchinin, Apr 06 2016 */
(GAP) List([0..30], n>Binomial(2*n+1, n+1)); # Muniru A Asiru, Feb 26 2019


CROSSREFS

Cf. A000110, A007318, A030662, A046097, A060897A060900, A049027, A076025, A076026, A060150, A001263, A005773, A001405, A132813, A134285.
Equals A000984(n+1)/2.
a(n) = (2*n+1)*Catalan(n) [A000108] = A035324(n+1, 1) (first column of triangle).
Row sums of triangles A028364, A050166, A039598.
Bisections: a(2*k) = A002458(k), a(2*k+1) = A001448(k+1)/2, k >= 0.
Other versions of the same sequence: A088218, A110556, A138364.
Diagonals 1 and 2 of triangle A100257.
Second row of array A102539.
Column of array A073165.
Row sums of A103371.  Susanne Wienand, Oct 22 2011
Cf. A002054: C(2*n+1, n1).  Bruno Berselli, Jan 20 2014
Cf. A005043, A091867, A001792, A001813, A166454.
Sequence in context: A167403 A318117 A110556 * A088218 A300975 A072266
Adjacent sequences: A001697 A001698 A001699 * A001701 A001702 A001703


KEYWORD

easy,nonn,nice,core


AUTHOR

N. J. A. Sloane


EXTENSIONS

Name corrected by Paul S. Coombes, Jan 11 2012
Name corrected by Robert Tanniru, Feb 01 2014


STATUS

approved



