A151471 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2 n steps taken from {(-1, -1), (-1, 0), (-1, 1), (1, -1), (1, 0), (1, 1)}.

1, 5, 70, 1335, 29722, 726726, 18924180, 515622393, 14537063970, 420926828322, 12451967363004, 374889010764450, 11453678709964092, 354319982375393300, 11078755353095301720, 349637888810064314295, 11124472891130365079250, 356502250246211571438090

Offset: 0

Author

Manuel Kauers, Nov 18 2008

Keywords

• nonn
• walk

Links

• M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, ArXiv 0810.4387, 2008.

Programs

Maple

G := Int(Int(2*((12*t^2+1)*hypergeom([1/4, 3/4],[1],64*t^2/(12*t^2+1)^2)
-12*t^2*hypergeom([3/4,5/4],[2],64*t^2/(12*t^2+1)^2))/((1-36*t^2)*(12*t^2+1)^(3/2)),t),t)/t^2;
ogf := subs(t=x^(1/2), series(G, t=0, 40)); # Mark van Hoeij, Aug 20 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[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[0, k, 2 n], {k, 0, 2 n}], {n, 0, 25}]

Formula

a(n) = ((1-n)*A026945(n)+(n+2)*A026945(n+1)) * (n+2)*binomial(2*n+3,n+1)/(3*(4*n+2)*(4*n+5)). [Mark van Hoeij, Jul 14 2010]