OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
FORMULA
G.f.: Int((hypergeom([-1/4,1/4],[1],64*x^2)-6*x*hypergeom([1/4,3/4],[2],64*x^2))/(1-8*x),x)/x. - Mark van Hoeij, Aug 20 2014
From Benedict W. J. Irwin, Oct 14 2016: (Start)
a(n) = Catalan(n) * binomial(n, floor(n/2)).
G.f.: 3F2(1/4,1/2,3/4; 1,3/2; 64*x^2] + (1 - 2F1(-1/4,1/4; 1; 64*x^2))/(4*x). (End)
D-finite with recurrence n*(n+1)^2*a(n) -4*n*(2*n-1)*a(n-1) -16*(n-1)*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Feb 08 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, -1 + j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]
Table[CatalanNumber[n]*Binomial[n, Floor[n/2]], {n, 0, 25}] (* G. C. Greubel, Oct 18 2016 *)
PROG
(PARI) a(n)=binomial(2*n, n) * binomial(n, n\2) / (n+1) \\ Charles R Greathouse IV, Oct 18 2016
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved