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%I #7 Dec 27 2023 21:30:52
%S 1,2,10,74,636,5996,60500,640536,7032004,79442404,918460996,
%T 10822792116,129586488144,1572865007248,19316089471728,
%U 239655258634016,3000258698957516,37860881282339972,481184329951150276,6154697345463251892,79178658464829291656,1023964603651763072264
%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)}.
%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, -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
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