A151312 - OEIS

%I #26 Aug 06 2024 04:54:50

%S 1,2,10,39,210,960,5340,26250,148610,761796,4360356,22971102,

%T 132469260,711426144,4124979144,22471772895,130833575730,720642479700,

%U 4209268283220,23389011317958,136971640993044,766537623135648,4498668129173400,25325478605719656,148897823229533196,842422760884567800

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

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

%H Robert Israel, <a href="/A151312/b151312.txt">Table of n, a(n) for n = 0..600</a>

%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 Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien <a href="https://doi.org/10.1016/j.ejc.2016.10.010">Hypergeometric expressions for generating functions of walks with small steps in the quarter plane.</a> Eur. J. Comb.  61, 242-275 (2017)

%H M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, <a href="http://arxiv.org/abs/0810.4387">ArXiv 0810.4387</a>.

%H A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, <a href="http://arxiv.org/abs/0811.2899">ArXiv 0811.2899</a>.

%F G.f.: Int(hypergeom([3/2,3/2],[2],16*x/((2*x+1)*(6*x+1)))*(1-2*x)/((2*x+1)*(6*x+1))^(3/2),x)/x. - _Mark van Hoeij_, Aug 14 2014

%p Gf:= Int(hypergeom([3/2,3/2],[2],16*x/((2*x+1)*(6*x+1)))*(1-2*x)/((2*x+1)*(6*x+1))^(3/2),x)/x:

%p S:= series(Gf,x,101):

%p seq(coeff(S,x,j),j=0..100); # _Robert Israel_, Aug 14 2014

%p # The following programs a0 and a1 give the even and odd terms:

%p a0 := proc(n) options remember;

%p   if n=0 then 1 elif n=1 then 10

%p   else  2*(-18*(2*n-3)^2*a0(n-2)+(2*n-1)*(20*n^2-2*n-3)*a0(n-1)/(2*n+1))/n^2

%p   fi

%p end:

%p a1 := proc(n) options remember;

%p   if n=0 then 2 elif n=1 then 39 else

%p    2*(-18*(2*n-1)^2*(1-1/n^2)*a1(n-2)+(20*n^2+18*n+1)*a1(n-1))/(n+1)^2

%p   fi

%p end:

%p seq(op([a0(n),a1(n)]),n=0..50); # - _Mark van Hoeij_, Jul 15 2022

%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[-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 0,2

%A _Manuel Kauers_, Nov 18 2008