%I #8 Dec 04 2016 13:57:03
%S 1,3,23,242,2987,40663,591642,9033815,143086147,2332577417,
%T 38920508426,661997451570,11442777854792,200524118645298,
%U 3555829498671923,63707928582145553,1151818635756842038,20992564017982373175,385356783440628359148,7119667287556648713060,132307647760313085491973
%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, 0), (1, 1)}.
%H Alois P. Heinz, <a href="/A151433/b151433.txt">Table of n, a(n) for n = 0..200</a>
%H M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, <a href="http://arxiv.org/abs/0810.4387">ArXiv 0810.4387</a>.
%p b:= proc(n, l) b(n, l):= `if`(l[1]<0 or l[2]<0 or n<l[1], 0, `if`(n=0, 1,
%p add(b(n-1, l+d), d=[[-1, -1], [-1, 0], [1, -1], [1, 0], [1, 1]])))
%p end:
%p a:= n-> b(2*n, [0$2]):
%p seq (a(n), n=0..30); # _Alois P. Heinz_, Jul 19 2012
%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, 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