%I #7 Jan 01 2024 02:35:03
%S 1,1,4,15,61,269,1245,5935,29128,145871,743421,3843154,20108453,
%T 106304763,567011174,3047948601,16496553378,89828010025,491790403030,
%U 2705562067688,14949910812086,82936556715572,461770124064327,2579548304347117,14453779059407019,81214726367783785,457521215541829301
%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, 0), (1, 1), (0, 1), (0, 1), (1, 1), (1, 1)}.
%H M. BousquetMÃ©lou 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[i, 1 + j, 1 + n] + aux[i, 1 + j, 1 + n] + aux[1 + i, 1 + j, 1 + n] + aux[1 + i, 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
