A151440 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of n steps taken from {(-1, -1), (-1, 1), (0, 1), (1, -1), (1, 1)}.

1, 1, 3, 8, 30, 104, 418, 1619, 6811, 27995, 121479, 518165, 2298497, 10057101, 45342297, 202127226, 922674852, 4172053826, 19231818364, 87941092870, 408591525432, 1885356748718, 8816855295648, 40987943515892, 192726338718178, 901557225420892, 4258807073126730, 20028060258669392

Offset

0,3

Links

Table of n, a(n) for n=0..27.

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

Formula

G.f.: Int(Int(2+Int(12*(1-x)*(1-2*Int((1-2*x-15*x^2)^(3/2)*((16*x^2+1)*(4680*x^6-1400*x^5-185*x^4-521*x^3+123*x^2+9*x-2)*hypergeom([9/4, 11/4],[3],64*x^2*(1+x^2)/(16*x^2+1)^2)-6*(-180*x^6+644*x^5+458*x^4-1622*x^3 +356*x^2-26*x-2)*x^2*hypergeom([11/4, 13/4],[4],64*x^2*(1+x^2)/(16*x^2+1)^2))/((16*x^2+1)^(11/2)*(1-x)^2),x))/(1-2*x-15*x^2)^(5/2),x),x),x)/((1+x)*x^2). - Mark van Hoeij, Aug 27 2014

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[i, -1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[0, k, n], {k, 0, n}], {n, 0, 25}]

Crossrefs

Sequence in context: A148887 A221537 A148888 * A213860 A360991 A361135

Adjacent sequences: A151437 A151438 A151439 * A151441 A151442 A151443

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved