A151318 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 1), (-1, 0), (0, 1), (1, 0), (1, 1)}.
1, 3, 13, 55, 249, 1131, 5253, 24543, 115825, 549331, 2620029, 12543367, 60270697, 290423355, 1403088885, 6793370415, 32956254945, 160152588195, 779470975725, 3798948989655, 18538237315545, 90565618791435, 442899758973285, 2167985089576575, 10621425660150609, 52078139149834611, 255533719072119133
Offset: 0
Links
- Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.
- Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
- Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, >arXiv:0810.4387 [math.CO], 2008.
- Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, >arXiv:0811.2899 [math.CO], 2008.
- Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 99.
Programs
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Mathematica
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[i, -1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]