A151423 - OEIS

%I #16 Dec 04 2016 13:57:02

%S 1,3,28,355,5264,85764,1488432,27030861,507976040,9804514720,

%T 193339562208,3880220133244,79026982569976,1629698960355600,

%U 33969388149210240,714666181953790035,15158444163422689080,323839596100845917400,6962822068346268247200,150567286583848676406480

%N 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, 1)}.

%H M. Bousquet-Mélou and M. Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, ArXiv 0810.4387, 2008.

%p G := Int(Int((2*(1-8*t^2)*hypergeom([3/4, 5/4],[1],64*t^2*(t^2+1)/(16*t^2+1)^2)

%p +12*t^2*hypergeom([3/4,5/4],[2],64*t^2*(t^2+1)/(16*t^2+1)^2))/(16*t^2+1)^(5/2),t),t)/t^2;

%p ogf := subs(t=x^(1/2), series(G, t=0, 40));  # _Mark van Hoeij_, Aug 20 2014

%t 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[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}]

%K nonn,walk

%O 0,2

%A _Manuel Kauers_, Nov 18 2008