A151493 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, 1, 6, 20, 114, 529, 3050, 16333, 96291, 554489, 3343913, 20074658, 123474818, 761154695, 4758617798, 29876386844, 189275072084, 1204491672779, 7713591678563, 49602001416920, 320478996972260, 2078111745252647, 13525786891484513, 88309928176783985, 578342399955533489, 3797738127548969927

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)*(3-Int(4*(1-2*x-35*x^2)^(3/2)*((12*x^2+1)*(-2520*x^6+420*x^5+930*x^4-225*x^3+385*x^2+25*x-5)*hypergeom([9/4, 11/4],[3],64*(x^2+x+1)*x^2/(12*x^2+1)^2)+6*(280*x^6-980*x^5+1830*x^4 +1875*x^3-640*x^2+15*x+5)*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)*(1-x)^2),x))/(1-2*x-35*x^2)^(5/2),x),x),x)/(x^2*(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: A207819 A297974 A280187 * A036755 A045470 A117998

Adjacent sequences: A151490 A151491 A151492 * A151494 A151495 A151496

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved