OFFSET
0,3
LINKS
M. Alekseyev, Count the number of words in a set, MathOverflow, 2018.
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.
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)
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
a(n) = Sum_{j=0..[n/2]} Sum_{i=max(j,[(n+1)/2]-j)..n-j} (i-j+1)*(2*(i+j)-n+1)/(i+j+1) * n!/(i+1)!/j!/(n-i-j)!. - Max Alekseyev, Mar 06 2018
G.f.: -(1/(x-1))*(1/2+1/x*Int(x^2/(x+1)^(1/2)/(1-3*x)^(3/2)*(13/2+Int(((1-3*x)/(x+1))^(1/2)*((10-1/x^3)*hypergeom([1/4, 3/4],[1],64*x^4)+6*(8*x^2+1)*(10*x^2-3)*hypergeom([5/4,7/4],[2],64*x^4)),x)),x)). - Mark van Hoeij, Oct 13 2009
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, 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}]
PROG
(PARI) { A151266(n) = sum(j=0, n\2, sum(i=max(j, (n+1)\2-j), n-j, (i-j+1)*(2*(i+j)-n+1)/(i+j+1) * n!/(i+1)!/j!/(n-i-j)! )); } \\ Max Alekseyev, Mar 06 2018
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved