OFFSET
0,3
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..500
Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, On sequences associated to the invariant theory of rank two simple Lie algebras, arXiv:1911.10288 [math.CO], 2019.
Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, On some combinatorial sequences associated to invariant theory, arXiv:2110.13753 [math.CO], 2021.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
FORMULA
G.f.: ((2*x+1)*(hypergeom([-2/3, -1/3],[1],27*x^2*(2*x+1))+4*x*hypergeom([-1/3, 1/3],[2],27*x^2*(2*x+1)))/(3*x+1)-1-3*x-5*x^2)/(3*x^3). - Mark van Hoeij, Aug 17 2014
Conjecture: (n+4)*(n+3)*a(n) -n*(n-1)*a(n-1) -12*(2*n+1)*(n-1)*a(n-2) -36*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jul 21 2017
G.f.: (-5*x^2-3*x-1)/(3*x^3)+(1+3*x)^3*hypergeom([1/4, 3/4],[2],64*x*(1+3*x)^3/(1+24*x+36*x^2)^2)/(3*x^3*(1+24*x+36*x^2)^(1/2)). - Mark van Hoeij, Jul 27 2021
MATHEMATICA
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, 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]]; Table[aux[0, 0, n], {n, 0, 25}]
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved