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

%I #7 Jan 01 2024 02:43:07

%S 1,1,3,8,25,85,291,1072,3949,15143,58563,231111,922494,3724920,

%T 15196256,62498021,259151683,1081560013,4542825176,19184568461,

%U 81432735810,347241072470,1486943756676,6392073797253,27576360268882,119363438165113,518246264053763,2256543142841418,9851632109402192

%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 n steps taken from {(-1, -1), (-1, 0), (0, -1), (0, 1), (1, 0)}.

%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>.

%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, j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[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}]

%K nonn,walk

%O 0,3

%A _Manuel Kauers_, Nov 18 2008