A002426 -id:A002426 - cp's OEIS frontend

A126869 a(n) = Sum_{k = 0..n} binomial(n,floor(k/2))*(-1)^(n-k).

1, 0, 2, 0, 6, 0, 20, 0, 70, 0, 252, 0, 924, 0, 3432, 0, 12870, 0, 48620, 0, 184756, 0, 705432, 0, 2704156, 0, 10400600, 0, 40116600, 0, 155117520, 0, 601080390, 0, 2333606220, 0, 9075135300, 0, 35345263800, 0, 137846528820, 0, 538257874440, 0, 2104098963720, 0, 8233430727600, 0, 32247603683100, 0, 126410606437752, 0

Offset: 0

Philippe Deléham, Mar 16 2007

Hankel transform is 2^n. Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.
From Andrew V. Sutherland, Feb 29 2008: (Start)
Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}.
Moment sequence of the trace of a random matrix in G = SO(2). If X = tr(A) is a random variable (A distributed with Haar measure on G), then a(n) = E[X^n].
Also the moment sequence of the trace of the k-th power of a random matrix in USp(2) = SU(2), for all k > 2.
(End)
From Paul Barry, Aug 10 2009: (Start)
The Hankel transform of 0,1,0,2,0,6,... is 0,-1,0,4,0,-16,0,... with general term I*(-4)^(n/2)(1 - (-1)^n)/4, I = sqrt(-1).
The Hankel transform of 1,1,0,2,0,6,... (which has g.f. 1 + x/sqrt(1 - 4*x^2)) is A164111. (End)
a(n) = A204293(2*n,n): central terms of the triangle in A204293. - Reinhard Zumkeller, Jan 14 2012
a(n) is the total number of closed walks (round trips) of length n on the graph P_N (a line with N nodes and N-1 edges), divided by N, in the limit N -> infinity. See a comment on A198632 and a link under A201198. - Wolfdieter Lang, Oct 10 2012

			a(4) = 6 {UUDD,UDUD,UDDU,DUUD,DUDU,DDUU}.

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011.
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 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012
Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 99.

This is A000984 with interspersed zeros. m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).

Cf. A107430, A061554, A126120.

a126869 n = a204293_row (2*n) !! n  -- Reinhard Zumkeller, Jan 14 2012

seq((-1)^(n/2)*pochhammer(-n,n/2)/(n/2)!, n=0..43); # Peter Luschny, May 17 2013
seq(n!*coeff(series(hypergeom([],[1],x^2),x,n+1),x,n),n=0..42); # Peter Luschny, Jan 31 2015

Table[(-1)^Floor[n/2] HypergeometricPFQ[{-n,-n},{1},-1],{n,0,30}] (* Peter Luschny, Nov 01 2011 *)

A126869 = lambda n: (2^(n-1)*((-1)^n+1)*gamma((n+1)/2))/(sqrt(pi)*gamma((n+2)/2))
[A126869(n) for n in range(44)] # Peter Luschny, Sep 10 2014

From Andrew V. Sutherland, Feb 29 2008: (Start)
a(2*n) = binomial(2*n,n) = A000984(n); a(2*n+1) = 0.
a(n) = Sum_{k = 0..n} A107430(n,k)*(-1)^(n-k).
a(n) = Sum_{k = 0..n} A061554(n,k)*(-1)^k.
a(n) = (1/Pi)*Integral_{t = 0..Pi} cos^n(t) dt. (End)
E.g.f.: I_0(2*x) where I_n(x) is the modified Bessel function as a function of x. - Benjamin Phillabaum, Mar 10 2011
G.f.: A(x) = 1/sqrt(1 - 4*x^2). - Vladimir Kruchinin, Apr 16 2011
a(n) = (1/Pi)*Integral{x = -2..2} x^n/sqrt((2 - x)*(2 + x)). - Peter Luschny, Sep 12 2011
a(n) = (-1)^floor(n/2) * Hypergeometric([-n,-n],[1], -1). - Peter Luschny, Nov 01 2011
E.g.f.: E(0)/(1 - x) where E(k) = 1 - x/(1 - x/(x - (k+1)^2/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013
E.g.f.: 1 + x^2/(Q(0) - x^2), where Q(k) = x^2 + (k+1)^2 - x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 28 2013
G.f.: 1/(1 - 2*x^2*Q(0)), where Q(k) = 1 + (4*k+1)*x^2/(k+1 - x^2*(2*k+2)*(4*k+3)/(2*x^2*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + (k+1)/(x*(2*k+1))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: G(0)/(1+x), where G(k) = 1 + x*(2+5*x)*(4*k+1)/((4*k+2)*(1+x)^2 - 2*(2*k+1)*(4*k+3)*x*(2+5*x)*(1+x)^2/((4*k+3)*x*(2+5*x) + 4*(k+1)*(1+x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 19 2014
a(n) = 2^n*JacobiP(n,0,-1/2-n,-3). - Peter Luschny, Aug 02 2014
a(n) = (2^(n-1)*((-1)^n+1)*Gamma((n+1)/2))/(sqrt(Pi)*Gamma((n+2)/2)). - Peter Luschny, Sep 10 2014
a(n) = n!*[x^n]hypergeom([],[1],x^2). - Peter Luschny, Jan 31 2015
a(n) = 2^n*hypergeom([1/2,-n],[1],2). - Peter Luschny, Feb 03 2015
From Peter Bala, Jul 25 2016: (Start)
a(n) = (-1)^floor(n/2)*Sum_{k = 0..n} (-1)^k*binomial(n,k)^2.
D-finite with recurrence: a(n) = 4*(n - 1)/n * a(n-2) with a(0) = 1, a(1) = 0. (End)
From Ilya Gutkovskiy, Jul 25 2016: (Start)
Inverse binomial transform of A002426.
a(n) = Sum_{k=0..n} (-1)^k*A128014(k).
a(n) ~ 2^n*((-1)^n + 1)/sqrt(2*Pi*n). (End)

A026375 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2k,k).

Original entry on oeis.org

1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605, 23973456915, 116312293305, 565280386625, 2751474553575, 13411044301945, 65448142561035, 319756851757695

Offset: 0

Clark Kimberling

nonn

a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=0; also a(n)=T(2n,n).
Partial sums of A085362. Number of bilateral Schroeder paths (i.e., lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at odd (positive or negative) levels. Example: a(2)=11 because we have HUD, UDH, UDUD, UUDD, UDDU, their reflections in the x-axis and HH. - Emeric Deutsch, Jan 30 2004
Largest coefficient of (1+3*x+x^2)^n; row sums of triangle in A124733. - Philippe Deléham, Oct 02 2007
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in three colors. - N-E. Fahssi, Feb 05 2008
Equals INVERT transform of A109033: (1, 2, 6, 22, 88, ...), INVERTi transform of A111966, binomial transform of A000984, and inverse Binomial transform of A081671. Convolved with A002212: (1, 3, 10, 36, ...) = A026376: (1, 6, 30, 144, ...). Equals convolution square root of A003463: (1, 6, 31, 156, 781, 3906, ...). - Gary W. Adamson, May 17 2009
Diagonal of array with rational generating function 1/(1 - (x^2 + 3*x*y + y^2)). - Gheorghe Coserea, Jul 29 2018
a(n) == 0 (mod 3) if and only if n is in A081606. - Fabio Visonà, Aug 03 2023

			G.f. = 1 + 3*x + 11*x^2 + 45*x^3 + 195*x^4 + 873*x^5 + 3989*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
David Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4.
Shu-Chiuan Chang and Robert Shrock, Structure of the Partition Function and transfer matrices for the Potts model in a magnetic field on lattice strips, J. Stat. Phys. 137 (2009) 667.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Rui Duarte and António Guedes de Oliveira, Generating functions of lattice paths, Univ. do Porto (Portugal 2023).
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012.
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 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Mathematics Stack Exchange, Proof that a(n) == 0 (mod 3) if and only if n is in A081606.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.
H. D. Nguyen and D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.

Column 3 of A292627. Column 1 of A110165. Central column of A272866.
Cf. A002893, A084610, A000172, A002212.
First differences are in A085362. Bisection of A026380.
m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).

List([0..25],n->Sum([0..n],k->Binomial(n,k)*Binomial(2*k,k))); # Muniru A Asiru, Jul 29 2018

a026375 n = a026374 (2 * n) n  -- Reinhard Zumkeller, Feb 22 2014

seq( add(binomial(n,k)*binomial(2*k,k), k=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001
a := n -> simplify(GegenbauerC(n, -n, -3/2)):
seq(a(n), n=0..22); # Peter Luschny, May 09 2016

Table[SeriesCoefficient[1/Sqrt[1-6*x+5*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
(* From Michael Somos, May 11 2014: (Start) *)
a[ n_] := Sum[ Binomial[n, k] Binomial[2 k, k], {k, 0, n}];
a[ n_] := If[ n < 0, 0, Hypergeometric2F1[-n, 1/2, 1, -4]];
a[ n_] := If[ n < 0, 0, Coefficient[(1 + 3 x + x^2)^n, x, n]];
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[Exp[3 x] BesselI[0,2 x], {x, 0, n}]];
(* (End) *)

A026375(n):=coeff(expand((1+3*x+x^2)^n),x,n);
makelist(A026375(n),n,0,12); /* Emanuele Munarini, Mar 02 2011 */

{a(n) = if( n<0, 0, polcoeff( (1 + 3*x + x^2)^n, n))}; /* Michael Somos, Sep 09 2002 */

a(n)={my(v=Vec((1-x-x^2)^n)); sum(k=1,#v, v[k]^2);} \\ Joerg Arndt, Jul 06 2011

{a(n) = sum(k=0, n, 5^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

{a(n) = sum(k=0, n\2, 3^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

Representation by Gauss's hypergeometric function, in Maple notation: a(n)=hypergeom([ -n, 1/2 ], [ 1 ], -4). - Karol A. Penson, Apr 20 2001
This sequence is the binomial transform of A000984. - John W. Layman, Aug 11 2000; proved by Emeric Deutsch, Oct 26 2002
E.g.f.: exp(3*x)*I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 17 2002
G.f.: 1/sqrt(1-6*x+5*x^2). - Emeric Deutsch, Oct 26 2002
D-finite with recurrence: n*a(n)-3*(2*n-1)*a(n-1)+5*(n-1)*a(n-2)=0 for n > 1. - Emeric Deutsch, Jan 24 2004
From Emeric Deutsch, Jan 30 2004: (Start)
a(n) = [t^n](1+3*t+t^2)^n;
a(n) = Sum_{j=ceiling(n/2)..n} 3^(2*j-n)*binomial(n, j)*binomial(j, n-j). (End)
a(n) = A026380(2*n-1) (n>0). - Emeric Deutsch, Feb 18 2004
G.f.: 1/(1-x-2*x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x... (continued fraction). - Paul Barry, Jan 06 2009
a(n) = sum of squared coefficients of (1+x-x^2)^n - see triangle A084610. - Paul D. Hanna, Jul 18 2009
a(n) = sum of squares of coefficients of (1-x-x^2)^n. - Joerg Arndt, Jul 06 2011
a(n) = (1/Pi)*Integral_{x=-2..2} ((3+x)^n/sqrt((2-x)*(2+x))) dx. - Peter Luschny, Sep 12 2011
a(n) ~ 5^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
G.f.: G(0)/(1-x), where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
0 = a(n)*(+25*a(n+1) - 45*a(n+2) + 10*a(n+3)) + a(n+1)*(-15*a(n+1) + 36*a(n+2) - 9*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, May 11 2014
a(n) = GegenbauerC(n, -n, -3/2). - Peter Luschny, May 09 2016
a(n) = Sum_{k=0..n} 5^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019
a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
a(n) = (1/Pi) * Integral_{x = -1..1} (1 + 4*x^2)^n/sqrt(1 - x^2) dx = (1/Pi) * Integral_{x = -1..1} (5 - 4*x^2)^n/sqrt(1 - x^2) dx. - Peter Bala, Jan 27 2020
From Peter Bala, Jan 10 2022: (Start)
1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 3*x^2 + 10*x^3 + 36*x^4 + ... is the o.g.f. of A002212.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

Definition simplified by N. J. A. Sloane, Feb 16 2012

A051286 Whitney number of level n of the lattice of the ideals of the fence of order 2n.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 63, 153, 376, 931, 2317, 5794, 14545, 36631, 92512, 234205, 594169, 1510192, 3844787, 9802895, 25027296, 63972861, 163701327, 419316330, 1075049011, 2758543201, 7083830648, 18204064403, 46812088751, 120452857976

Offset: 0

Emanuele Munarini

A Chebyshev transform of the central trinomial numbers A002426: image of 1/sqrt(1-2x-3x^2) under the mapping that takes g(x) to (1/(1+x^2))*g(x/(1+x^2)). - Paul Barry, Jan 31 2005
a(n) has same parity as Fibonacci(n+1) = A000045(n+1); see A107597. - Paul D. Hanna, May 22 2005
This is the second kind of Whitney numbers, which count elements, not to be confused with the first kind, which sum Mobius functions. - Thomas Zaslavsky, May 07 2008
From Paul Barry, Mar 31 2010: (Start)
Apply the Riordan array (1/(1-x+x^2),x/(1-x+x^2)) to the aerated central binomial coefficients with g.f. 1/sqrt(1-4x^2).
Hankel transform is A174882. (End)
a(n) is the number of lattice paths in L[n]. The members of L[n] are lattice paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step h with weight 1, an (1,0)-step H with weight 2, a (1,1)-step U with weight 2, and a (1,-1)-step D with weight 1. The weight of a path is the sum of the weights of its steps. Example: a(3)=5 because we have hhh, hH, Hh, UD, and DU; a(4)=11 because we have hhhh, hhH, hHh, Hhh, HH, hUD, UhD, UDh, hDU, DhU, and DUh (see the Bona-Knopfmacher reference).
Apparently the number of peakless grand Motzkin paths of length n. - David Scambler, Jul 04 2013
A bijection between L[n] (as defined above) and peakless grand Motzkin paths of length n is now given in arXiv:2002.12874. - Sergi Elizalde, Jul 14 2021
a(n) is also the number of unimodal bargraphs with a centered maximum (i.e., whose column heights are weakly increasing in the left half and weakly decreasing in the right half) and semiperimeter n+1. - Sergi Elizalde, Jul 14 2021
Diagonal of the rational function 1 / ((1 - x)*(1 - y) - (x*y)^2). - Ilya Gutkovskiy, Apr 23 2025
a(n) is the number of rooted ordered trees with node weights summing to n, where the root has weight 0, non-root node weights are in {1,2}, and no nodes have the same weight as their parent node. - John Tyler Rascoe, Jun 10 2025

			a(3) = 5 because the ideals of size 3 of the fence F(6) = { x1 < x2 > x3 < x4 > x5 < x6 } are x1*x3*x5, x1*x2*x3, x3*x4*x5, x1*x5*x6, x3*x5*x6.

Vincenzo Librandi and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects, in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018.
Andrei Asinowski, Axel Bacher, Cyril Banderier and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
Cyril Banderier and Paweł Hitczenko, Enumeration and asymptotics of restricted compositions having the same number of parts, Disc. Appl. Math. 160 (18) (2012) 2542-2554. See Puzzle 3.1.
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Miklós Bóna and Arnold Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
Steffen Eger, The Combinatorics of Weighted Vector Compositions, arXiv:1704.04964 [math.CO], 2017.
Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2021.
Edyta Hetmaniok, Barbara Smoleń and Roman Wituła, The Stirling triangles, Proceedings of the Symposium for Young Scientists in Technology, Engineering and Mathematics (SYSTEM 2017), Kaunas, Lithuania, April 28, 2017, p. 35-41.
Ivo L. Hofacker, Christian M. Reidys, and Peter F. Stadler, Symmetric circular matchings and RNA folding. Discr. Math., 312:100-112, 2012. See Prop. 5, C_l^{1}(z).
Emanuele Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Jesús Sistos Barrón and Hua Wang, Balanced n-color compositions, Integers (2024) Vol. 24A, Art. No. A2. See pp. 4, 5, 8, 14.

Cf. main diagonal of A125250, column k=2 of A384747.

Cf. A051291, A051292, A078698, A107597, A185828 (log), A174882 (Hankel transf.).

seq( sum('binomial(i-k,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1$2, 2, 5][n+1],
     ((2*n-1)*a(n-1)+(n-1)*a(n-2)+(2*n-3)*a(n-3)-(n-2)*a(n-4))/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 11 2016

Table[Sum[Binomial[n-k,k]^2,{k,0,Floor[n/2]}],{n,0,40}] (* Emanuele Munarini, Mar 01 2011; corrected by Harvey P. Dale, Sep 12 2012 *)
CoefficientList[Series[1/Sqrt[1-2*x-x^2-2*x^3+x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 05 2013 *)
a[n_] := HypergeometricPFQ[ {(1-n)/2, (1-n)/2, -n/2, -n/2}, {1, -n, -n}, 16]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Feb 26 2013 *)

makelist(sum(binomial(n-k,k)^2,k,0,floor(n/2)),n,0,40);  /* Emanuele Munarini, Mar 01 2011 */

a(n)=polcoeff(1/sqrt((1+x+x^2)*(1-3*x+x^2)+x*O(x^n)),n)

a(n)=sum(k=0,n,binomial(n-k,k)^2) /* Paul D. Hanna */

{a(n)=polcoeff( exp(sum(m=1,n, sum(k=0,m, binomial(2*m,2*k)*x^k) *x^m/m) +x*O(x^n)), n)}  /* Paul D. Hanna, Mar 18 2011 */

{a(n)=local(A=1); A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n, x^m*sum(k=0, n, binomial(m+k, k)^2*x^k) * (1-x)^(2*m+1) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n\2, x^(2*m) * sum(k=0, n, binomial(m+k, k)^2*x^k) +x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

{a(n)=local(A=1+x); A=sum(m=0, n\2, x^(2*m) * sum(k=0, m, binomial(m, k)^2*x^k) / (1-x +x*O(x^n))^(2*m+1) ); polcoeff(A, n)} \\ Paul D. Hanna, Sep 05 2014

from sympy import binomial
def a(n): return sum(binomial(n - k, k)**2 for k in range(n//2 + 1))
print([a(n) for n in range(31)]) # Indranil Ghosh, Apr 18 2017

G.f.: 1/sqrt(1 - 2*x - x^2 - 2*x^3 + x^4).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A002426(n-2k). - Paul Barry, Jan 31 2005
From Paul D. Hanna, May 22 2005: (Start)
a(n) = Sum_{k=0..n} C(n-k, k)^2.
Limit_{n->oo} a(n+1)/a(n) = (sqrt(5)+3)/2.
G.f.: 1/sqrt((1+x+x^2)*(1-3*x+x^2)). (End)
a(n) = Sum_{k=0..n} A049310(n, k)^2. - Philippe Deléham, Nov 21 2005
a(n) = Sum_{k=0..n} (C(k,k/2)*(1+(-1)^k)/2) * Sum_{j=0..n} (-1)^((n-j)/2)*C((n+j)/2,j)*((1+(-1)^(n-j))/2)*C(j,k). - Paul Barry, Mar 31 2010
G.f.: exp( Sum_{n>=1} (x^n/n)*Sum_{k=0..n} C(2n,2k)*x^k ). - Paul D. Hanna, Mar 18 2011
Logarithmic derivative equals A185828. - Paul D. Hanna, Mar 18 2011
D-finite with recurrence: n*a(n) - (2*n-1)*a(n-1) - (n-1)*a(n-2) - (2*n-3)*a(n-3) + (n-2)*a(n-4) = 0. - R. J. Mathar, Dec 17 2011
The g.f. A(x) satisfies the differential equation (1-2*x-x^2-2*x^3+x^4)*A'(x) = (1+x+3*x^2-2*x^3)*A(x), from which the recurrence conjectured by Mathar follows. - Emanuele Munarini, Dec 18 2017
a(n) ~ phi^(2*n + 2) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 05 2013, simplified Dec 18 2017
From Paul D. Hanna, Sep 05 2014: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * x^k.
G.f.: Sum_{n>=0} x^n *[Sum_{k>=0} C(n+k,k)^2 * x^k] * (1-x)^(2*n+1).
G.f.: Sum_{n>=0} x^(2*n) * [Sum_{k>=0} C(n+k,k)^2 * x^k].
G.f.: Sum_{n>=0} x^(2*n) * [Sum_{k=0..n} C(n,k)^2 * x^k] /(1-x)^(2n+1).
(End)

A006139 na(n) = 2(2n-1)a(n-1) + 4(n-1)a(n-2) with a(0) = 1.

Original entry on oeis.org

1, 2, 8, 32, 136, 592, 2624, 11776, 53344, 243392, 1116928, 5149696, 23835904, 110690816, 515483648, 2406449152, 11258054144, 52767312896, 247736643584, 1164829376512, 5484233814016, 25852072517632, 121997903495168

Offset: 0

N. J. A. Sloane

a(n) = number of Delannoy paths (A001850) from (0,0) to (n,n) in which every Northeast step is immediately preceded by an East step. - David Callan, Mar 14 2004
The Hankel transform (see A001906 for definition) of this sequence is A036442 : 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005
In general, 1/sqrt(1-4*r*x-4*r*x^2) has e.g.f. exp(2rx)BesselI(0,2r*sqrt((r+1)/r)x), a(n) = Sum_{k=0..n} C(2k,k)*C(k,n-k)*r^k, gives the central coefficient of (1+(2r)x+r(r+1)x^2) and is the (2r)-th binomial transform of 1/sqrt(1-8*C(n+1,2)x^2). - Paul Barry, Apr 28 2005
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H and U steps can have two colors. - N-E. Fahssi, Feb 05 2008
Self-convolution of a(n)/2^n gives Pell numbers A000129(n+1). - Vladimir Reshetnikov, Oct 10 2016
This sequence gives the integer part of an integral approximation to Pi, and also appears in Frits Beukers's "A Rational Approach to Pi" (cf. Links, Example). Despite quality M ~ 0.9058... reported by Beukers, measurements between n = 10000 and 30000 lead to a contentious quality estimate, M ~ 0.79..., at the 99% confidence level. In "Searching for Apéry-Style Miracles" Doron Zeilberger Quotes that M = 0.79119792... and also gives a closed form. The same rational approximation to Pi also follows from time integration on a quartic Hamiltonian surface, 2*H=(q^2+p^2)*(1-4*q*(q-p)). - Bradley Klee, Jul 19 2018, updated Mar 17 2019
Diagonal of rational function 1/(1 - (x + y + x*y^2)). - Gheorghe Coserea, Aug 06 2018

			G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 136*x^4 + 592*x^5 + 2624*x^6 + 11776*x^7 + ...
J_3 = Integral_{y=0..Pi/4} 4*(4*(sin(y)-cos(y))*sin(y))^3*dy = 32*Pi - (304/3), |J_3| < 1. - _Bradley Klee_, Jul 19 2018

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

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
Frits Beukers, A rational approach to Pi, Nieuw archief voor wiskunde 5/1 No. 4, December 2000, p. 377.
Dario Castellanos, A generalization of Binet's formula and some of its consequences, Fib. Quart., 27 (1989), 424-438.
Maciej Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv:1410.5747 [math.CO], 2014.
Shalosh B. Ekhad and Doron Zeilberger, Searching for Apéry-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm], arXiv:1405.4445 [math.NT], 2014.
Bradley Klee, Approximating Pi with Trigonometric-Polynomial Integrals, Wolfram Demonstrations, July 27, 2018.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Cf. A001850, A002426, A036442, A084600-A084606, A084608-A084615.
Cf. A106258, A106259, A106260, A106261.
First column of A110446. A higher-quality Pi approximation: A123178.

a:=[1,2];; for n in [3..25] do a[n]:=1/(n-1)*(2*(2*n-3)*a[n-1]+4*(n-2)*a[n-2]); od; a; # Muniru A Asiru, Aug 06 2018

seq(add(binomial(2*k, k)*binomial(k, n-k), k=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001
A006139 := n -> 2^n*hypergeom([-n/2, 1/2-n/2], [1], 2):
seq(simplify(A006139(n)), n=0..29); # Peter Luschny, Sep 18 2014

Table[SeriesCoefficient[1/(1-4x-4x^2)^(1/2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 05 2012 *)
Table[Abs[LegendreP[n, I]] 2^n, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
Table[Sum[Binomial[2*k, k]*Binomial[k, n - k], {k,0,n}], {n,0,50}] (* G. C. Greubel, Feb 28 2017 *)
a[n_] := If[n == 0, 1, Coefficient[(1 + 2 x + 2 x^2)^n, x^n]] (* Emanuele Munarini, Aug 04 2017 *)
CoefficientList[Series[1/Sqrt[(-4 x^2 - 4 x + 1)], {x, 0, 24}], x] (* Robert G. Wilson v, Jul 28 2018 *)

a(n) := coeff(expand((1+2*x+2*x^2)^n),x,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Aug 04 2017 */

for(n=0,30,t=polcoeff((1+2*x+2*x^2)^n,n,x); print1(t","))

for(n=0,25, print1(sum(k=0,n, binomial(2*k,k)*binomial(k,n-k)), ", ")) \\ G. C. Greubel, Feb 28 2017

{a(n) = (-2*I)^n * pollegendre(n, I)}; /* Michael Somos, Aug 04 2018 */

a(n) = Sum_{k=0..n} C(2*k, k)*C(k, n-k). - Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001
G.f.: 1/(1-4x-4x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+2x^2)^n. - Paul D. Hanna, Jun 01 2003
Inverse binomial transform of central Delannoy numbers A001850. - David Callan, Mar 14 2004
E.g.f.: exp(2*x) * BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004
a(n) = Sum_{k=0..floor(n/2)} C(n,2k) * C(2k,k) * 2^(n-k). - Paul Barry, Sep 19 2006
a(n) ~ 2^(n - 3/4) * (1 + sqrt(2))^(n + 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Oct 05 2012, simplified Jan 31 2023
G.f.: 1/(1 - 2*x*(1+x)*Q(0)), where Q(k)= 1 + (4*k+1)*x*(1+x)/(k+1 - x*(1+x)*(2*k+2)*(4*k+3)/(2*x*(1+x)*(4*k+3)+(2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
a(n) = 2^n*hypergeom([-n/2, 1/2-n/2], [1], 2). - Peter Luschny, Sep 18 2014
0 = a(n)*(+16*a(n+1) + 24*a(n+2) - 8*a(n+3)) + a(n+1)*(+8*a(n+1) + 16*a(n+2) - 6*a(n+3)) + a(n+2)*(-2*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Oct 13 2016
It appears that Pi/2 = Sum_{n >= 1} (-1)^(n-1)*4^n/(n*a(n-1)*a(n)). - Peter Bala, Feb 20 2017
G.f.: G(x) = (1/(2*Pi))*Integral_{y=0..2*Pi} 1/(1-x*(4*(sin(y)-cos(y))*sin(y)))*dy, also satisfies: (2+4*x)*G(x)-(1-4*x-4*x^2)*G'(x)=0. - Bradley Klee, Jul 19 2018
a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit. - Seiichi Manyama, Aug 29 2025

A005717 Construct triangle in which n-th row is obtained by expanding (1 + x + x^2)^n and take the next-to-central column.

Original entry on oeis.org

1, 2, 6, 16, 45, 126, 357, 1016, 2907, 8350, 24068, 69576, 201643, 585690, 1704510, 4969152, 14508939, 42422022, 124191258, 363985680, 1067892399, 3136046298, 9217554129, 27114249960, 79818194925, 235128465026, 693085098852, 2044217638456, 6032675068061

Offset: 1

N. J. A. Sloane

Number of ordered trees with n+1 edges, having root of even degree and nonroot nodes of outdegree at most 2. - Emeric Deutsch, Aug 02 2002
The connection to Motzkin numbers comes from the Lagrange inversion formula. - Michael Somos, Oct 10 2003
Number of horizontal steps in all Motzkin paths of length n. - Emeric Deutsch, Nov 09 2003
Number of UHD's in all Motzkin paths of length n+2 (here U=(1,1), H=(1,0) and D=(1,-1)). Example: a(2)=2 because in the nine Motzkin paths of length 4, HHHH, HHUD, HUDH, H(UHD), UDHH, UDUD, (UHD)H, UHHD and UUDD, we have altogether two UHD's (shown between parentheses). - Emeric Deutsch, Dec 26 2003
Number of ordered trees with n+1 edges, having exactly one leaf at even height. Number of Dyck path of semilength n+1, having exactly one peak at even height. Example: a(3)=6 because we have uuu(ud)ddd, u(ud)dudud, udu(ud)dud, ududu(ud)d, u(ud)uuddd and uuudd(ud)d (here u=(1,1),d=(1,-1) and the unique peak at even height is shown between parentheses). - Emeric Deutsch, Mar 10 2004
a(n) is the number of Dyck (n+1)-paths containing exactly one UDU. - David Callan, Jul 15 2004
Number of peaks in all Motzkin paths of length n+1. - Emeric Deutsch, Sep 01 2004
This is a kind of Motzkin transform of A059841 because the substitution x -> x*A001006(x) in the independent variable of the g.f. of A059841 generates 1,0,1,2,6,16,... that is 1,0 followed by this sequence here. - R. J. Mathar, Nov 08 2008
a(n) is the number of lattice paths avoiding N^(>=3) from (0,0) to (n,n). - Shanzhen Gao, Apr 20 2010
a(n+1) is the number of binary strings having n 0's and n 1's and no appearance of 000.  For example, for n = 1, there 2 strings: 01 and 10.  For n = 2, there are 6: 0011, 0101, 0110, 1001, 1010, 1100. - Toby Gottfried, Sep 12 2011
a(n) is the number of paths in the half-plane x>=0, from (0,0) to (n,1), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=3, we have the 6 paths HHU, HUH, UDU, UUD, UHH, DUU. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) is the number of ways to tile a strip of length 2*n+1 with squares, dominos, and trominos, where the number of trominos is always one more than the number of squares. - Greg Dresden and Anna Kalynchuk, Jul 30 2025

			G.f. = x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 126*x^6 + 357*x^7 + ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..1000 (Terms 1 to 200 computed by T. D. Noe; terms 201 to 1000 by G. C. Greubel, Jan 15 2017)
Kassie Archer and Christina Graves, A new statistic on Dyck paths for counting 3-dimensional Catalan words, arXiv:2205.09686 [math.CO], 2022.
Jean-Luc Baril, Sergey Kirgizov, José L. Ramírez, and Diego Villamizar, The Combinatorics of Motzkin Polyominoes, arXiv:2401.06228 [math.CO], 2024. See page 13.
Emeric Deutsch, Ordered trees with prescribed root degrees, node degrees, and branch lengths, Discrete Mathematics 282 (2004), 89-94.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Richard K. Guy, Letter to N. J. A. Sloane, 1987.
Stanislav Krymski and Alexander Okhotin, Longer Shortest Strings in Two-Way Finite Automata, in: Jirásková G., Pighizzini G. (eds) Descriptional Complexity of Formal Systems. DCFS 2020. Lecture Notes in Computer Science, vol 12442. Springer, Cham.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, arXiv:0912.0072 [math.NT], 2009; FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
Mark Shattuck, Subword Patterns in Smooth Words, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 6.
Chenying Wang, Piotr Miska, and István Mező, The r-derangement numbers, Discrete Mathematics 340(7) (2017), 1681-1692.
Eric Weisstein's World of Mathematics, Trinomial Coefficient.

A diagonal of A027907.

Cf. A001006, A002426, A005043, A005773, A076540 (binomial transform).

seq(add(binomial(i, k) *binomial(i-k, k+1), k=0..floor(i/2)), i=1..30); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001
M:= proc(n) option remember; `if` (n<2, 1, (3*(n-1)*M(n-2) +(2*n+1) *M(n-1))/ (n+2)) end: A005717 := n -> n*M(n-1):
seq(A005717(i), i=1..27); # Peter Luschny, Sep 12 2011
a := n -> simplify(GegenbauerC(n,-n-1,-1/2)):
seq(a(n), n=0..28); # Peter Luschny, May 07 2016

Table[Coefficient[Expand[(1+x+x^2)^n], x, n-1], {n, 1, 40}]
Table[n*Hypergeometric2F1[(1 - n)/2, 1 - n/2, 2, 4], {n, 29}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
Table[GegenbauerC[n,-n-1,-1/2],{n,0,100}] (* Emanuele Munarini, Oct 20 2016 *)

makelist(ultraspherical(n,-n-1,-1/2),n,0,12); /* Emanuele Munarini, Oct 20 2016 */

{a(n) = if( n<0, 0, polcoeff( (1 + x + x^2)^n, n-1))}; /* Michael Somos, Sep 09 2002 */

{a(n) = if( n<0, 0, n * polcoeff( serreverse( x / (1 + x + x^2) + x * O(x^n)), n))}; /* Michael Somos, Oct 10 2003 */

N=10^3;  x='x+'x*O('x^N);
gf = 2*x/(1-2*x-3*x^2+(1-x)*sqrt(1-2*x-3*x^2));
v005717 = Vec(gf);
/* Joerg Arndt, Aug 16 2012 */

def A():
    a, b, n = 0, 1, 1
    while True:
        yield b
        n += 1
        a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))
A005717 = A()
print([next(A005717) for  in range(29)]) # _Peter Luschny, May 16 2016

a(n) = Sum_{k=1..n} T(k, k-1), where T is the array defined in A025177.
G.f.: 2*x/(1-2*x-3*x^2+(1-x)*sqrt(1-2*x-3*x^2)). - Emeric Deutsch, Aug 14 2002
E.g.f.: exp(x) * I_1(2x), where I_1 is the Bessel function. - Michael Somos, Sep 09 2002
a(n) = A111808(n,n-1). - Reinhard Zumkeller, Aug 17 2005
a(n) = Sum_{k=0..floor((n-1)/3)} (-1)^k * binomial(n,k) * binomial(2n-2-3k, n-1). - David Callan, Jul 03 2006
From Paul Barry, Feb 05 2007: (Start)
a(n) = n*Sum_{k=0..floor((n-1)/2), C(n-1,2k)*C(k)}, C(n) = A000108(n).
a(n) = Sum_{k=0..floor((n-1)/2)} (2k+1)*C(n,2k+1)*C(k).
a(n) = Sum_{k=0..n-1} ( Sum_{j=0..floor(k/2)} C(k,2j)*C(2j+1,j) ). (End)
a(n) = (A002426(n+1) - A002426(n))/2. - Paul Barry, May 22 2008
a(n) = n*A001006(n-1). - Paul Barry, Oct 05 2009
a(n) = Sum_{i=0..floor(n/2)} C(n+1,n-i) * C(n-i,i). - Shanzhen Gao, Apr 20 2010
D-finite with recurrence: (n+1)*a(n) - 3*n*a(n-1) - (n+3)*a(n-2) + 3*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 28 2011
a(n) ~ 3^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2013
0 = a(n) * 3*(n+1)*(n+2) + a(n+1) * (n+2)*(2*n+3) - a(n+2) * (n+1)*(n+3) for all n in Z. - Michael Somos, Apr 03 2014
G.f.: z*M(z)/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 19 2015
Working with an offset of 0, a(n) = [x^n](1 + x + x^2)^(n+1); binomial transform is A076540. - Peter Bala, Jun 15 2015
a(n) = GegenbauerC(n,-n-1,-1/2). - Peter Luschny, May 07 2016
a(n) = (-1)^(n+1) * n * hypergeom([3/2, 1-n], [3], 4). - Vladimir Reshetnikov, Sep 28 2016
a(n) = Sum_{k=0..n-1} binomial(n,k)*binomial(n-k, k+1) [Krymski and Okhotin]. - Michel Marcus, Dec 04 2020
a(n) = (1/2)*(A005773(n+1) - A005043(n)). - Peter Bala, Feb 11 2022
a(n) = A002426(n) - A005043(n). - Amiram Eldar, May 17 2024

More terms from Erich Friedman, Jun 01 2001

A051924 a(n) = binomial(2n,n) - binomial(2n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

Original entry on oeis.org

1, 4, 14, 50, 182, 672, 2508, 9438, 35750, 136136, 520676, 1998724, 7696444, 29716000, 115000920, 445962870, 1732525830, 6741529080, 26270128500, 102501265020, 400411345620, 1565841089280, 6129331763880, 24014172955500, 94163002754652, 369507926510352

Offset: 1

Barry E. Williams, Dec 19 1999

Number of partitions with Ferrers plots that fit inside an n X n box, but not in an n-1 X n-1 box. - Wouter Meeussen, Dec 10 2001
From Benoit Cloitre, Jan 29 2002: (Start)
Let m(1,j)=j, m(i,1)=i and m(i,j) = m(i-1,j) + m(i,j-1); then a(n) = m(n,n):
  1  2  3  4 ...
  2  4  7 11 ...
  3  7 14 25 ...
  4 11 25 50 ... (End)
This sequence also gives the number of clusters and non-crossing partitions of type D_n. - F. Chapoton, Jan 31 2005
If Y is a 2-subset of a 2n-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
Prefaced with a 1: (1, 1, 4, 14, 50, ...) and convolved with the Catalan sequence = A097613: (1, 2, 7, 25, 91, ...). - Gary W. Adamson, May 15 2009
Total number of up steps before the second return in all Dyck n-paths. - David Scambler, Aug 21 2012
Conjecture: a(n) mod n^2 = n+2 iff n is an odd prime. - Gary Detlefs, Feb 19 2013
First differences of A000984 and A030662. - J. M. Bergot, Jun 22 2013
From R. J. Mathar, Jun 30 2013: (Start)
Equivalent to the Meeussen comment and the Bergot comment: The array view of A007318 is
  1,   1,   1,   1,   1,   1,
  1,   2,   3,   4,   5,   6,
  1,   3,   6,  10,  15,  21,
  1,   4,  10,  20,  35,  56,
  1,   5,  15,  35,  70, 126,
  1,   6,  21,  56, 126, 252,
and a(n) are the hook sums Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
From Gus Wiseman, Apr 12 2019: (Start)
Equivalent to Wouter Meeussen's comment, a(n) is the number of integer partitions (of any positive integer) such that the maximum of the length and the largest part is n. For example, the a(1) = 1 through a(3) = 14 partitions are:
  (1)  (2)   (3)
       (11)  (31)
       (21)  (32)
       (22)  (33)
             (111)
             (211)
             (221)
             (222)
             (311)
             (321)
             (322)
             (331)
             (332)
             (333)
(End)
Coxeter-Catalan numbers for Coxeter groups of type D_n [Armstrong]. - N. J. A. Sloane, Mar 09 2022
a(n+1) is the number of ways that a best of n pairs contest with early termination can go. For example, the first stage of an association football (soccer) penalty-kick shoot out has n=5 pairs of shots and there are a(6)=672 distinct ways it can go. For n=2 pairs, writing G for goal and M for miss, and listing the up-to-four shots in chronological order with teams alternating shots, the n(3)=14 possibilities are MMMM, MMMG, MMGM, MMGG, MGM, MGGM, MGGG, GMMM, GMMG, GMG, GGMM, GGMG, GGGM, and GGGG. Not all four shots are taken in two cases because it becomes impossible for one team to overcome the lead of the other team. - Lee A. Newberg, Jul 20 2024

			Sums of {1}, {2, 1, 1}, {2, 2, 3, 3, 2, 1, 1}, {2, 2, 4, 5, 7, 6, 7, 5, 5, 3, 2, 1, 1}, ...

Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274 16; See Table 2.8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, arXiv:math/0611106 [math.CO], 2006-2007.
Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See p. 10.
F. Chapoton, Clusters.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.
Sergey Fomin and Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Milan Janjic, Two Enumerative Functions.
Joshua Marsh and Nathan Williams, Nesting Nonpartitions, J. Int. Seq., Vol. 25 (2022), Article 22.8.8.
Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig, Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian, arXiv:1910.14048 [cond-mat.str-el], 2019.
Hugh Thomas, Tamari Lattices and Non-Crossing Partitions in Types B and D, arXiv:math/0311334 [math.CO], 2003-2005.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study, Vol. 56, No. 1 (2023), 1-17.

Left-central elements of the (1, 2)-Pascal triangle A029635.
Column sums of A096771.
Cf. A000108, A024482 (diagonal from 2), A076540 (diagonal from 3), A000124 (row from 2), A004006 (row from 3), A006522 (row from 4).
Cf. A128064; first differences of A000984.
Cf. A097613.
Cf. A115720, A252464, A257990, A263297, A325189, A325192, A325193.

a051924 n = a051924_list !! (n-1)
a051924_list = zipWith (-) (tail a000984_list) a000984_list
-- Reinhard Zumkeller, May 25 2013

[Binomial(2*n, n)-Binomial(2*n-2, n-1): n in [1..28]]; // Vincenzo Librandi, Dec 21 2016

C:= n-> binomial(2*n, n)/(n+1): seq((n+1)*C(n)-n*C(n-1), n=1..25); # Emeric Deutsch, Jan 08 2008
Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..24); # Zerinvary Lajos, Jan 01 2007
a := n -> 2^(-2+2*n)*GAMMA(-1/2+n)*(3*n-2)/(sqrt(Pi)*GAMMA(1+n)):
seq(simplify(a(n)), n=1..24); # Peter Luschny, Dec 14 2015

Table[Binomial[2n,n]-Binomial[2n-2,n-1],{n,30}] (* Harvey P. Dale, Jan 15 2012 *)

a(n)=binomial(2*n,n)-binomial(2*n-2,n-1) \\ Charles R Greathouse IV, Jun 25 2013

{a(n)=polcoeff((1-x) / sqrt(1-4*x +x*O(x^n)) - 1,n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 08 2014

{a(n)=polcoeff( sum(m=1, n, x^m * sum(k=0, m, binomial(m, k)^2 * x^k) / (1-x +x*O(x^n))^(2*m)), n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 08 2014

a = lambda n: 2^(-2+2*n)*gamma(n-1/2)*(3*n-2)/(sqrt(pi)*gamma(1+n))
[a(n) for n in (1..120)] # Peter Luschny, Dec 14 2015

G.f.: (1-x) / sqrt(1-4*x) - 1. - Paul D. Hanna, Nov 08 2014
G.f.: Sum_{n>=1} x^n/(1-x)^(2*n) * Sum_{k=0..n} C(n,k)^2 * x^k. - Paul D. Hanna, Nov 08 2014
a(n+1) = binomial(2*n, n) + 2*Sum_{i=0..n-1} binomial(n+i, i) (V's in Pascal's Triangle). - Jon Perry Apr 13 2004
a(n) = n*C(n-1) - (n-1)*C(n-2), where C(n) = A000108(n) = Catalan(n). For example, a(5) = 50 = 5*C(4) - 4*C(3) - 5*14 - 3*5 = 70 - 20. Triangle A128064 as an infinite lower triangular matrix * A000108 = A051924 prefaced with a 1: (1, 1, 4, 14, 50, 182, ...). - Gary W. Adamson, May 15 2009
Sum of 3 central terms of Pascal's triangle: 2*C(2+2*n, n)+C(2+2*n, 1+n). - Zerinvary Lajos, Dec 20 2005
a(n+1) = A051597(2n,n). - Philippe Deléham, Nov 26 2006
The sequence 1,1,4,... has a(n) = C(2*n,n)-C(2*(n-1),n-1) = 0^n+Sum_{k=0..n} C(n-1,k-1)*A002426(k), and g.f. given by (1-x)/(1-2*x-2*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-.... (continued fraction). - Paul Barry, Oct 17 2009
a(n) = (3*n-2)*(2*n-2)!/(n*(n-1)!^2) = A001700(n) + A001791(n-1). - David Scambler, Aug 21 2012
D-finite with recurrence: a(n) = 2*(3*n-2)*(2*n-3)*a(n-1)/(n*(3*n-5)). - Alois P. Heinz, Apr 25 2014
a(n) = 2^(-2+2*n)*Gamma(-1/2+n)*(3*n-2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) ~ (3/4)*4^n*(1-(7/24)/n-(7/128)/n^2-(85/3072)/n^3-(581/32768)/n^4-(2611/262144)/n^5)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
E.g.f.: ((1 - x)*BesselI(0,2*x) + x*BesselI(1,2*x))*exp(2*x) - 1. - Ilya Gutkovskiy, Dec 20 2016
a(n) = 2 * A097613(n) for n > 1. - Bruce J. Nicholson, Jan 06 2019
Sum_{n>=1} a(n)/8^n = 7/(4*sqrt(2)) - 1. - Amiram Eldar, May 06 2023

Edited by N. J. A. Sloane, May 03 2008, at the suggestion of R. J. Mathar

A074940 Numbers having at least one 2 in their ternary representation.

Original entry on oeis.org

2, 5, 6, 7, 8, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 32, 33, 34, 35, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 83, 86, 87, 88, 89, 92

Offset: 1

Benoit Cloitre and Reinhard Zumkeller, Oct 04 2002; revised Dec 03 2003

Also, numbers m such that 3 divides C(2m,m).
Also, numbers m such that the central trinomial coefficient A002426(m) == 0 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
Also, numbers m such that A092255(m) == 0 (mod 3). - Benoit Cloitre, Mar 22 2004
Also, numbers m such that the coefficient of x^m equals 0 in Product_{k>=0} (1-x^(3^k)). - N. J. A. Sloane, Jun 01 2010

			12 is not in the sequence since it is 110_3, but 11 is in the sequence since it is 102_3. - _Michael B. Porter_, Jun 30 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

Complement of A005836.
Cf. A006996, A007089, A081603, A081610, A081605, A081606.
A039966(a(n)) = 0.

a074940 n = a074940_list !! (n-1)
a074940_list = filter ((== 0) . a039966) [0..]
-- Reinhard Zumkeller, Jun 06 2012, Sep 29 2011

Select[Range@ 120, MemberQ[IntegerDigits[#, 3], 2] &] (* or *)
Select[Range@ 120, Divisible[Binomial[2 #, #], 3] &] (* Michael De Vlieger, Jun 29 2016 *)
Select[Range[100],DigitCount[#,3,2]>0&] (* Harvey P. Dale, Aug 25 2019 *)

is(n)=while(n,if(n%3==2,return(1));n\=3);0 \\ Charles R Greathouse IV, Aug 21 2011

from gmpy2 import digits
def A074940(n):
    def f(x):
        s = digits(x,3)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return n+int(s,2)
        return n+int(s[:i]+'1'*(l-i),2)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 29 2024

a(n) = n + O(n^0.631). - Charles R Greathouse IV, Aug 21 2011

More terms from Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

A124696 Number of base-3 circular n-digit numbers with adjacent digits differing by 1 or less.

Original entry on oeis.org

1, 3, 7, 15, 35, 83, 199, 479, 1155, 2787, 6727, 16239, 39203, 94643, 228487, 551615, 1331715, 3215043, 7761799, 18738639, 45239075, 109216787, 263672647, 636562079, 1536796803, 3710155683, 8957108167, 21624372015, 52205852195

Offset: 0

R. H. Hardin, Dec 28 2006

These are the number of smooth cyclic words of length n over the alphabet {1,2,3}. See theorem 3.3 in Knopfmacher and others. - Peter Luschny, Aug 13 2012

This is the main entry for 234 similar sequences. Cf. the link to the OEIS Wiki for a list, the programs and a derivation of the linear recurrences. - Georg Fischer, Apr 09 2021

R. H. Hardin, Table of n, a(n) for n = 0..210 [uploaded by Georg Fischer, Apr 05 2021]
Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, and Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], 2024. See p. 5.
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Cf. A002426, Row 3 of A276562.

T := (n, k) -> `if`(n=0, 1, add((1 + 2*cos(j*Pi/(k + 1)))^n, j=1..k)):
a := n -> simplify(T(n, 3)): seq(a(n), n=0..28); # Peter Luschny, Mar 28 2021

[Empirical] a(base,n) = a(base-1,n) + A002426(n+1) for base = 1..floor(n/2)+1.
a(n) = T(n,3) for n > 0, where T(n,k) = Sum_{j=1..k} (1 + 2*cos(j*Pi/(k + 1)))^n. - Peter Luschny, Aug 13 2012
From Colin Barker, Nov 26 2012: (Start)
a(n) = 1 + (1 - sqrt(2))^n + (1 + sqrt(2))^n for n > 0.
a(n) = 3*a(n-1) - a(n-2) - a(n-3) for n > 3.
G.f.: -(2*x^3 + x^2 - 1)/((x - 1)*(x^2 + 2*x - 1)). (End)
a(n) = A002203(n)+1, n>0. - R. J. Mathar, May 09 2023

A039966 a(0) = 1; thereafter a(3n+2) = 0, a(3n) = a(3n+1) = a(n).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Dec 11 1999

nonn

Number of partitions of n into distinct powers of 3.
Trajectory of 1 under the morphism: 1 -> 110, 0 -> 000. Thus 1 -> 110 ->110110000 -> 110110000110110000000000000 -> ... - Philippe Deléham, Jul 09 2005
Also, an example of a d-perfect sequence.
This is a composite of two earlier sequences contributed at different times by N. J. A. Sloane and by Reinhard Zumkeller, Mar 05 2005. Christian G. Bower extended them and found that they agreed for at least 512 terms. The proof that they were identical was found by Ralf Stephan, Jun 13 2005, based on the fact that they were both 3-regular sequences.

			The triples of elements (a(3k), a(3k+1), a(3k+2)) are (1,1,0) if a(k) = 1 and (0,0,0) if a(k) = 0.  So since a(2) = 0, a(6) = a(7) = a(8) = 0, and since a(3) = 1, a(9) = a(10) = 1 and a(11) = 0. - _Michael B. Porter_, Jul 11 2016

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences that are fixed points of mappings
Index entries for characteristic functions

For generating functions Product_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
Cf. A062051, A000009, A000244, A004642.
Characteristic function of A005836 (and apart from offset of A003278).

a039966 n = fromEnum (n < 2 || m < 2 && a039966 n' == 1)
   where (n',m) = divMod n 3
-- Reinhard Zumkeller, Sep 29 2011

a := proc(n) option remember; if n <= 1 then RETURN(1) end if; if n = 2 then RETURN(0) end if; if n mod 3 = 2 then RETURN(0) end if; if n mod 3 = 0 then RETURN(a(1/3*n)) end if; if n mod 3 = 1 then RETURN(a(1/3*n - 1/3)) end if end proc; # Ralf Stephan, Jun 13 2005

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) s = Rest[ Sort[ Plus @@@ Table[UnrankSubset[n, Table[3^i, {i, 0, 4}]], {n, 32}]]]; Table[ If[ Position[s, n] == {}, 0, 1], {n, 105}] (* Robert G. Wilson v, Jun 14 2005 *)
CoefficientList[Series[Product[(1 + x^(3^k)), {k, 0, 5}], {x, 0, 111}], x] (* or *)
Nest[ Flatten[ # /. {0 -> {0, 0, 0}, 1 -> {1, 1, 0}}] &, {1}, 5] (* Robert G. Wilson v, Mar 29 2006 *)
Nest[ Join[#, #, 0 #] &, {1}, 5] (* Robert G. Wilson v, Jul 27 2014 *)

{a(n)=local(A,m); if(n<0, 0, m=1; A=1+O(x); while(m<=n, m*=3; A=(1+x)*subst(A,x,x^3)); polcoeff(A,n))} /* Michael Somos, Jul 15 2005 */

A039966(n)=vecmax(digits(n+!n,3))<2;
apply(A039966, [0..99]) \\ M. F. Hasler, Feb 15 2023

def A039966(n):
    while n > 2:
        n,r = divmod(n,3)
        if r==2: return 0
    return int(n!=2) # M. F. Hasler, Feb 15 2023

a(0) = 1, a(1) = 0, a(n) = b(n-2), where b is the sequence defined by b(0) = 1, b(3n+2) = 0, b(3n) = b(3n+1) = b(n). - Ralf Stephan
a(n) = A005043(n-1) mod 3. - Christian G. Bower, Jun 12 2005
a(n) = A002426(n) mod 3. - John M. Campbell, Aug 24 2011
a(n) = A000275(n) mod 3. - John M. Campbell, Jul 08 2016
Properties: 0 <= a(n) <= 1, a(A074940(n)) = 0, a(A005836(n)) = 1; A104406(n) = Sum(a(k), 1 <= k <= n). - Reinhard Zumkeller, Mar 05 2005
Euler transform of sequence b(n) where b(3^k) = 1, b(2*3^k) = -1 and zero otherwise. - Michael Somos, Jul 15 2005
G.f. A(x) satisfies A(x) = (1+x)*A(x^3). - Michael Somos, Jul 15 2005
G.f.: Product{k>=0} 1+x^(3^k). Exponents give A005836.

Entry revised Jun 30 2005

Offset corrected by John M. Campbell, Aug 24 2011

A089942 Inverse binomial matrix applied to A039599.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 3, 2, 1, 3, 6, 6, 3, 1, 6, 15, 15, 10, 4, 1, 15, 36, 40, 29, 15, 5, 1, 36, 91, 105, 84, 49, 21, 6, 1, 91, 232, 280, 238, 154, 76, 28, 7, 1, 232, 603, 750, 672, 468, 258, 111, 36, 8, 1, 603, 1585, 2025, 1890, 1398, 837, 405, 155, 45, 9, 1, 1585, 4213, 5500

Offset: 0

Paul Barry, Nov 16 2003

Reverse of A071947 - related to lattice paths. First column is A005043.
Triangle T(n,k), 0 <= k <= n, defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Feb 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Riordan array (f(x),x*g(x)), where f(x)is the o.g.f. of A005043 and g(x)is the o.g.f. of A001006. - Philippe Deléham, Nov 22 2009
Riordan array ((1+x-sqrt(1-2x-3x^2))/(2x(1+x)), (1-x-sqrt(1-2x-3x^2))/(2x)). Inverse of Riordan array ((1+x)/(1+x+x^2),x/(1+x+x^2)). E.g.f. of column k is exp(x)*(Bessel_I(k,2x)-Bessel_I(k+1,2x)).
Diagonal sums are A187306.
Simultaneous equations using the first n rows solve for diagonal lengths of odd N = (2n+1) regular polygons, with constants c^0, c^1, c^2, ...; where c = 1 + 2*cos( 2*Pi/N) = sin(3*Pi/N)/sin(Pi/N) = the third longest diagonal of N>5.  By way of example, take the first 4 rows relating to the 9-gon (nonagon), N=(2*4 + 1), with c = 1 + 2*cos(2*Pi/9) = 2.5320888.... The simultaneous equations are (1,0,0,0) = 1; (0,1,0,0) = c; (1,1,1,0) = c^2, (1,3,2,1) = c^3. The answers are 1, 2.532..., 2.879..., and 1.879...; the four distinct diagonal lengths of the 9-gon (nonagon) with edge = 1. - Gary W. Adamson, Sep 07 2011
Number of linearly independent irreducible representations of a given weight j in a tensor of given rank n. - Mikkel N. Schmidt, Aug 20 2025

			Triangle begins
   1,
   0,   1,
   1,   1,   1,
   1,   3,   2,   1,
   3,   6,   6,   3,   1,
   6,  15,  15,  10,   4,  1,
  15,  36,  40,  29,  15,  5,  1,
  36,  91, 105,  84,  49, 21,  6, 1,
  91, 232, 280, 238, 154, 76, 28, 7, 1
Production matrix is
  0, 1,
  1, 1, 1,
  0, 1, 1, 1,
  0, 0, 1, 1, 1,
  0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 1, 1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paul Barry and Aoife Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
J. A. R. Coope, R. F. Snider, and F. R. McCourt, Irreducible Cartesian Tensors, The Journal of Chemical Physics, vol 43, no 7, 1965.
Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
Toufik Mansour and Mark Shattuck, Enumeration of Catalan and smooth words according to capacity, Integers (2025) Vol. 25, Art. No. A5. See p. 29.
Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.
Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths, Eur. J. Comb. (2014) Vol. 39, 157-169. See Table 2.2.

Row sums give A002426 (central trinomial coefficients).

T:= (n,k) -> simplify(GegenbauerC(n-k,-n+1,-1/2)-GegenbauerC(n-k-1,-n+1,-1/2)): for n from 1 to 9 do seq(T(n,k), k=1..n) od; # Peter Luschny, May 12 2016
# Or by recurrence:
T := proc(n, k) option remember;
if n = k then 1 elif k < 0 or n < 0 or k > n then 0
elif k = 0 then T(n-1, 1) else T(n-1, k-1) + T(n-1, k) + T(n-1, k+1) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, May 25 2021

T[n_, k_] := GegenbauerC[n - k, -n + 1, -1/2] - GegenbauerC[n - k - 1, -n + 1, -1/2]; Table[T[n, k], {n,1,10}, {k,1,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

G.f.: (1+z-q)/[(1+z)(2z-t+tz+tq)], where q = sqrt(1-2z-3z^2).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A005043(m+n). - Philippe Deléham, Mar 22 2007
Sum_{k=0..n} T(n,k)*(2k+1) = 3^n. - Philippe Deléham, Mar 22 2007
Sum_{k=0..n} T(n,k)*2^k = A112657(n). - Philippe Deléham, Apr 01 2007
T(n,2k) + T(n,2k+1) = A109195(n,k). - Philippe Deléham, Nov 11 2008
T(n,k) = GegenbauerC(n-k,-n+1,-1/2) - GegenbauerC(n-k-1,-n+1,-1/2) for 1 <= k <= n. - Peter Luschny, May 12 2016

Edited by Emeric Deutsch, Mar 04 2004

cp's OEIS Frontend

A126869 a(n) = Sum_{k = 0..n} binomial(n,floor(k/2))*(-1)^(n-k).

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A026375 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k).

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A051286 Whitney number of level n of the lattice of the ideals of the fence of order 2n.

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A006139 n*a(n) = 2*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2) with a(0) = 1.

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A005717 Construct triangle in which n-th row is obtained by expanding (1 + x + x^2)^n and take the next-to-central column.

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A026375 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2k,k).

A006139 na(n) = 2(2n-1)a(n-1) + 4(n-1)a(n-2) with a(0) = 1.

A051924 a(n) = binomial(2n,n) - binomial(2n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).