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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..25.

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: A240008 A151322 A002320 * A181662 A241478 A113140

Adjacent sequences:  A151320 A151321 A151322 * A151324 A151325 A151326

KEYWORD

nonn,walk

AUTHOR

Manuel Kauers, Nov 18 2008

STATUS

approved

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Last modified April 18 10:56 EDT 2021. Contains 343087 sequences. (Running on oeis4.)