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A151255
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)}
0
1, 1, 2, 3, 8, 15, 39, 77, 216, 459, 1265, 2739, 7842, 17641, 49854, 113175, 327604, 761787, 2182833, 5101595, 14868582, 35338401, 102146176, 243510453, 713019480, 1721265625, 5005198029, 12105626337, 35565979706, 86870058279, 253706973975, 620415879229, 1827423157812, 4504531840875, 13199126952109
OFFSET
0,3
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
A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.
A. Bostan and M. Kauers, 2008. Automatic Classification of Restricted Lattice Walks, ArXiv 0811.2899.
M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.
FORMULA
G.f. (1-2*x)*(1-((1-3*x)*(1+x))^(1/2)*(1-Int((2*(1-6*x^2-8*x^3)*hypergeom([1/4, 3/4],[1],64*x^4)+8*x^3*(1-7*x+4*x^2)*hypergeom([3/4, 5/4],[2],64*x^4))/((1-2*x)^2*((1-3*x)*(1+x))^(1/2)*(1+x)),x))/(1-3*x))/(4*x^2). - Mark van Hoeij, Aug 16 2014
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, j, -1 + n] + aux[1 + i, -1 + 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
Sequence in context: A055543 A308433 A049957 * A147999 A148000 A148001
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved