OFFSET
0,3
LINKS
A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899v2 [math.CO], 2008.
MAPLE
H := hypergeom([1/4, 7/12], [4/3], 64*x*(x-1)*(3*x+1)^2/(1-5*x)^4);
U := ((1816*x^5 - 3192*x^4 + 382*x^3 + 140*x^2 - 13*x - 1)*hypergeom([1/4, 3/4, 7/6], [1, 4/3], 256*x^4) + 3*(2*x-1)*(3*x+1)*(16*x^2+1)*(212*x^3 - 82*x^2 + 4*x + 1)*hypergeom([3/4, 7/6, 5/4], [1, 4/3], 256*x^4) - 16*x^3*(564*x^4 + 142*x^3 - 89*x^2 + 8*x - 1)*hypergeom([3/4, 7/6, 5/4], [4/3, 2], 256*x^4))/((5*x-1)*(2*x-1))^3;
Ord := 40; seriesInt := proc(A, x) global Ord; int(series(A, x, Ord), x) end:
S := seriesInt((1-5*x)*x^(-4/3)*(1+3*x)^(-5/3)*(1-5*x)^(4/3)*(1-x)^(-4/3)*H^(-2)*(1-seriesInt(H*U, x)), x):
ogf := series((1-2*x)/(4*x^2) + (1/12) * (1-2*x)*x^(-5/3)*(1+3*x)^(2/3)*(1-5*x)^(-4/3)*(1-x)^(1/3)*H*S, x, Ord);
# So the o.g.f. can be expressed in terms of 3F2 and 2F1 functions. - Mark van Hoeij, Apr 10 2012
MATHEMATICA
aux[i_Integer, j_Integer, k_Integer, n_Integer] := Which[Min[i, j, k, n] < 0 || Max[i, j, k] > n, 0, n == 0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1 + i, j, k, -1 + n] + aux[1 + i, -1 + j, -1 + k, -1 + n] + aux[1 + i, -1 + j, k, -1 + n] + aux[1 + i, 1 + j, k, -1 + n] + aux[1 + i, 1 + j, 1 + k, -1 + n]]; Table[Sum[aux[i, j, k, n], {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved