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A151323
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), (0, 1), (1, 0), (1, 1)}.
0
1, 3, 14, 67, 342, 1790, 9580, 52035, 285990, 1586298, 8864676, 49844238, 281719164, 1599314652, 9113895960, 52109150691, 298806189318, 1717855010274, 9898828072692, 57158263594458, 330662400729492, 1916134078427556, 11120825740970088, 64634042348169294, 376139362185133404, 2191569966890629380
OFFSET
0,2
LINKS
A. Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.
Alin Bostan, Calcul Formel pour la Combinatoire des Marches, Habilitation à Diriger des Recherches, Université Paris 13, December 2017.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
FORMULA
G.f. appears to be (((1+2*x)/(1-6*x))^(1/4)-1)/(2*x). [From Mark van Hoeij, Nov 20 2009]
Conjecture: (n+1)*a(n) -2*(2*n+1)*a(n-1) -12*(n-1)*a(n-2) = 0. - R. J. Mathar, Oct 26 2012
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[i, 1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]
CROSSREFS
Sequence in context: A351068 A345683 A002320 * A354503 A181662 A241478
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved