|
%I #21 Mar 25 2020 22:43:56
%S 1,3,14,67,342,1790,9580,52035,285990,1586298,8864676,49844238,
%T 281719164,1599314652,9113895960,52109150691,298806189318,
%U 1717855010274,9898828072692,57158263594458,330662400729492,1916134078427556,11120825740970088,64634042348169294,376139362185133404,2191569966890629380
%N 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)}.
%H A. Bostan, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.410.1160&rep=rep1&type=pdf">Computer Algebra for Lattice Path Combinatorics</a>, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
%H A. Bostan and M. Kauers, <a href="https://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008-2009.
%H Alin Bostan, <a href="https://specfun.inria.fr/bostan/HDR.pdf">Calcul Formel pour la Combinatoire des Marches</a>, Habilitation à Diriger des Recherches, Université Paris 13, December 2017.
%H Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, <a href="https://arxiv.org/abs/2001.00393">Stieltjes moment sequences for pattern-avoiding permutations</a>, arXiv:2001.00393 [math.CO], 2020.
%H M. Bousquet-Mélou and M. Mishna, <a href="http://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.
%F G.f. appears to be (((1+2*x)/(1-6*x))^(1/4)-1)/(2*x). [From _Mark van Hoeij_, Nov 20 2009]
%F 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
%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, 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}]
%K nonn,walk
%O 0,2
%A _Manuel Kauers_, Nov 18 2008
|
