OFFSET
0,2
LINKS
A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.
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.
A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
FORMULA
G.f.: Int(1+Int((2*x+1)*(6*x+1)*(6+Int(3*(1-4*x-12*x^2)^(3/2)*((8*x^2-1)*(96*x^3-248*x^2-150*x-13)*hypergeom([5/4, 7/4],[2],64*x^3*(2*x+1)/(8*x^2-1)^2)+5*(8*x^2+4*x+1)*(32*x^3-32*x^2-42*x-5)*hypergeom([7/4, 9/4],[2],64*x^3*(2*x+1)/(8*x^2-1)^2))/(2*(2*x+1)*(6*x+1)^2*(1-8*x^2)^(7/2)),x))/(1-4*x-12*x^2)^(5/2),x),x)/x. - Mark van Hoeij, Aug 16 2014
a(n) ~ [x^n]( (1-2*x)*(1+2*x)^(1/2)/(4*x*(1-6*x)^(1/2)) ), where [x^n](f) denotes the coefficient of x^n in the series expansion of f. - Mark van Hoeij, May 28 2020
a(n) ~ 2^(n+1) * 3^(n - 1/2) / sqrt(Pi*n) [Bostan and Kauers, p.13]. - Vaclav Kotesovec, May 29 2020
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[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, 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