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Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (0, -1), (1, 1)}.
3

%I #21 Aug 06 2024 04:48:49

%S 1,1,3,7,17,47,125,333,939,2597,7183,20505,57859,163201,469795,

%T 1341775,3830529,11092823,31940165,91927379,267406401,774447755,

%U 2242022721,6544458687,19036737381,55354815639,162028272261,472921269031,1379896701413,4048204328607,11848014062621,34665885291485

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

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 997.

%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 Alois P. Heinz, <a href="/A151265/b151265.txt">Table of n, a(n) for n = 0..1000</a>

%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, <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 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 a(n) == 1 (mod 2). - _Alois P. Heinz_, Jul 12 2021

%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[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}]

%Y Column k=2 of A335570.

%K nonn,walk

%O 0,3

%A _Manuel Kauers_, Nov 18 2008