A151284 - OEIS

%I #23 Jan 10 2024 04:59:58

%S 1,2,6,20,70,254,942,3550,13532,52030,201386,783560,3061442,12001804,

%T 47181278,185904220,733908634,2901998092,11490757796,45552262860,

%U 180762964146,717939220774,2853611232902,11349816190552,45168339253888,179845805435900,716409551285034,2854926106932244

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

%H M. Bousquet-Mélou and M. Mishna, <a href="https://arxiv.org/abs/0810.4387">Walks with small steps in the quarter plane</a>, arXiv:0810.4387 [math.CO], 2008-2009.

%H A. Bostan and M. Kauers, <a href="https://arxiv.org/abs/0811.2899">Automatic Classification of Restricted Lattice Walks</a>, arXiv:0811.2899 [math.CO], 2008-2009.

%H MathOverFlow, <a href="https://mathoverflow.net/a/322394/41291">Enumeration of lattice paths of a specific type</a>.

%H Kilian Raschel, <a href="https://doi.org/10.4171/jems/317">Counting walks in a quadrant: a unified approach via boundary value problems</a>, J. Eur. Math. Soc. (JEMS) 14, No. 3, 749-777 (2012).

%F G.f.: (3 + 2*(1 - 4*z^2)*u*Sum_{k>=1}((-1)^k*z^{k-1}*w^k/(1 + z^{k+1}*w^k*u)^2)/(1 - 4*z) where w = (1 - sqrt(1 - 4*z^2))/(2*z^2) = 1 + z^2 + 2*z^4 + 5*z^6 + 14*z^8 + ... and u = ((1 - 3*z)*(1 + z - z^2*w) - sqrt((1 - 2*z)*(1 - 2*z - 7*z^2)*(1 - z^2*w)))/(2*z^3) = 1 + 2*z + 7*z^2 + 20*z^3 + 66*z^4 + ... (see MathOverFlow link). - _Mamuka Jibladze_, Dec 24 2023

%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, j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[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}]

%K nonn,walk

%O 0,2

%A _Manuel Kauers_, Nov 18 2008