0,2

Alois P. Heinz, Table of n, a(n) for n = 0..500

A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d'Informatique de Paris Nord, Université Paris 13, December 2017.

Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, and Lucien Pech, Hypergeometric expressions for generating functions of walks with small steps in the quarter plane. Eur. J. Comb. 61, 242-275 (2017).

A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

Natalia L. Skirrow, chessboards

G.f.: (1/x)*Integral_{x} -((16*x^2+24*x-1) / (1+4*x)^5) * hypergeom([5/4, 5/4],[2],-2*x/(x+1/4)^4*(x+1)*(x-1/8)) dx. - Mark van Hoeij, Oct 13 2009

G.f.: (1/x) * Integral_{x} hypergeom([3/2,3/2],[2],16*x*(1+x)/(1+4*x)^2)/(1+4*x)^3. - Mark van Hoeij, Aug 14 2014

From Vaclav Kotesovec, Feb 13 2026: (Start)

Recurrence: n*(n+1)^2*(3*n - 1)*a(n) = n*(9*n^3 + 18*n^2 + n - 4)*a(n-1) + 4*(n-1)*(27*n^3 - 3*n^2 - 7*n + 2)*a(n-2) + 32*(n-2)*(n-1)*n*(3*n + 2)*a(n-3).

a(n) ~ 2^(3*n+3) / (3*Pi*n). (End)

From Natalia L. Skirrow, Feb 22 2026: (Start)

With ct = cos(Pi*th), cp = cos(Pi*ph), a(n) = Integral_{th,ph=0..1} (1+ct)*(1+cp) * ((1+2*ct)*(1+2*cp)-1)^n.

a(n) = Sum_{i,j=0..n, i+j>=n} C(n,n-i,n-j) * A001405(i) * A001405(j), where C(_,_,_) is a trinomial coefficient.

G.f.: Integral_{th=0..1} (1+ct)/(1+2*ct)*(sqrt{ (1+2*x*(1+ct)) / (1-2*x*(1+3*ct)) } - 1) / (2*x). (End)

From Peter Luschny, Mar 06 2026: (Start)

a(n) = Sum_{k=0..n} A107230(n, n - k) * A393138(n, k).

a(n) = Sum_{k=0..n} A393140(n, k).

a(p) is divisible by p if p is prime. (End)

T := proc(n, k) option remember; ifelse(k = 0, binomial(n, iquo(n, 2)), iquo((n - k + 1) * T(n, k - 1) + n * T(n - 1, k - 1), iquo((k + 1), 2))) end: a := n -> local k; add(T(n, k), k = 0..n): seq(a(n), n = 0.. 23); # Peter Luschny, Mar 06 2026

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]

CoefficientList[Series[Integrate[HypergeometricPFQ[{3/2, 3/2}, {2}, 16*x*(1+x)/(1+4*x)^2]/(1+4*x)^3, x]/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 16 2014, after Mark van Hoeij *)

(Python)

from math import comb

def A151331(n: int) -> int:

return sum(comb(n, i) * comb(i, i >> 1) *

sum(comb(i, n - j) * comb(j, j >> 1) for j in range(n - i, n + 1))

for i in range(n + 1) )

print([A151331(n) for n in range(24)]) # Natalia L. Skirrow, Feb 22 2026

Cf. A025595 (8 X 8 version), A107230, A393138, A393140.

Sequence in context: A391219 A007277 A025595 * A137962 A267662 A081341

Adjacent sequences: A151328 A151329 A151330 * A151332 A151333 A151334

nonn,walk

Manuel Kauers, Nov 18 2008

approved