OFFSET
0,2
REFERENCES
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
LINKS
Robert Israel, Table of n, a(n) for n = 0..600
A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.
Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien Hypergeometric expressions for generating functions of walks with small steps in the quarter plane. Eur. J. Comb. 61, 242-275 (2017)
M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.
A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.
FORMULA
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
MAPLE
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:
S:= series(Gf, x, 101):
seq(coeff(S, x, j), j=0..100); # Robert Israel, Aug 14 2014
# The following programs a0 and a1 give the even and odd terms:
a0 := proc(n) options remember;
if n=0 then 1 elif n=1 then 10
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
fi
end:
a1 := proc(n) options remember;
if n=0 then 2 elif n=1 then 39 else
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
fi
end:
seq(op([a0(n), a1(n)]), n=0..50); # - Mark van Hoeij, Jul 15 2022
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[-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}]
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved