%I #22 Aug 06 2024 04:56:32
%S 1,1,5,13,61,199,939,3389,16129,61601,295373,1169939,5643023,22931063,
%T 111114587,460242509,2238229577,9409963861,45894616065,195273012089,
%U 954636984885,4102002557665,20092702499037,87052952678691,427107841926791,1863539746958059,9155858788887359,40191781253867359
%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, -1), (1, 1)}.
%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.: Int(-1+Int((1+x)*(3*x+1)*Int(12*(1-2*x-15*x^2)^(3/2)*((8*x^2-1)*(2280*x^5+621*x^4-104*x^3-117*x^2-14*x+2)*hypergeom([7/4, 9/4],[2],64*(x^2+1)*x^2/(16*x^2+1)^2)-x*(280*x^6-297*x^5-562*x^4-406*x^3+128*x^2+11*x+10)*hypergeom([7/4, 9/4],[3],64*(x^2+1)*x^2/(16*x^2+1)^2))/((3*x+1)*(16*x^2+1)^(9/2)*(1+x)^2),x)/(1-2*x-15*x^2)^(5/2),x),x)/((x-1)*x). - _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, -1 + j, -1 + n] + aux[-1 + i, 1 + 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