A151492 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, 0), (-1, 1), (0, -1), (1, -1), (1, 0), (1, 1)}.

1, 0, 5, 5, 72, 158, 1503, 4958, 38390, 161170, 1116376, 5433892, 35471944, 189340318, 1199984953, 6790736977, 42491355502, 249738458994, 1557129072666, 9386156958774, 58598281352232, 359465513555718, 2252350134784800, 13992962318626614, 88084177345901544, 552491177540617338

Offset

0,3

Links

Table of n, a(n) for n=0..25.

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(6*(1-x)*(1-Int(4*(1-2*x-35*x^2)^(3/2)*((12*x^2+1)*(-10*x^6+289*x^5-515*x^4-1802*x^3-487*x^2+8*x+6)*hypergeom([9/4, 11/4],[3],64*(x^2+x+1)*x^2/(12*x^2+1)^2)+6*(590*x^6-571*x^5-103*x^4+3079*x^3 +769*x^2-14*x-6)*x^2*hypergeom([11/4, 13/4],[4],64*(x^2+x+1)*x^2/(12*x^2+1)^2))/((12*x^2+1)^(11/2)*(x-1)^2),x))/(1-2*x-35*x^2)^(5/2),x),x),x)/(x^2*(x-1)). - 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, 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, 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: A171726 A129358 A318420 * A081050 A081049 A048607

Adjacent sequences: A151489 A151490 A151491 * A151493 A151494 A151495

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved