A151261 - OEIS

%I #20 Aug 06 2024 04:55:59

%S 1,1,3,5,17,34,121,265,969,2246,8351,20118,75676,187524,711369,

%T 1801161,6876749,17710990,67962103,177474154,683797608,1806245948,

%U 6982719679,18624035846,72200420096,194169343564,754537816956,2043762871460,7958294450084,21691396049160,84615387463873,231907336128537

%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, 1), (-1, 0), (1, 0)}.

%H A. Bostan, <a href="https://citeseerx.ist.psu.edu/pdf/749aef4c6f3668e652b5074e5268346ccecc88c9">Computer Algebra for Lattice Path Combinatorics</a>, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

%H Alin Bostan, <a href="https://specfun.inria.fr/bostan/HDR.pdf">Calcul Formel pour la Combinatoire des Marches</a> [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017.

%H A. Bostan and M. Kauers, <a href="http://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008-2009.

%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.: (1-2*x)*(1+(-1+2*Int(((1-24*x^3)*hypergeom([1/2, 3/2],[2],16*x^2/(1+4*x^2))+18*x^2*(2*x-1)*hypergeom([1/2, 5/2],[3],16*x^2/(1+4*x^2)))/((1-4*x)^(1/2)*(1-2*x)^2*(1+4*x^2)^(1/2)),x))/(1-4*x)^(1/2))/(4*x^2). - _Mark van Hoeij_, Aug 16 2014

%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, j, -1 + n] + aux[1 + 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 0,3

%A _Manuel Kauers_, Nov 18 2008