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%I #7 Dec 27 2023 21:29:30
%S 1,1,2,5,11,32,87,248,764,2263,7125,22502,71308,232915,756571,2496821,
%T 8313892,27728911,93585460,316624455,1077272837,3687696937,
%U 12654416060,43655883878,151090181445,524542100001,1827895867960,6385078233779,22370743311415,78580314467710,276639299185638,976285257901368
%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), (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, 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