A151284 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)}.

1, 2, 6, 20, 70, 254, 942, 3550, 13532, 52030, 201386, 783560, 3061442, 12001804, 47181278, 185904220, 733908634, 2901998092, 11490757796, 45552262860, 180762964146, 717939220774, 2853611232902, 11349816190552, 45168339253888, 179845805435900, 716409551285034, 2854926106932244

Offset

0,2

Links

Table of n, a(n) for n=0..27.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.

A. Bostan and M. Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009.

MathOverFlow, Enumeration of lattice paths of a specific type.

Kilian Raschel, Counting walks in a quadrant: a unified approach via boundary value problems, J. Eur. Math. Soc. (JEMS) 14, No. 3, 749-777 (2012).

Formula

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

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[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: A087433 A119373 A360292 * A049138 A095929 A078482

Adjacent sequences: A151281 A151282 A151283 * A151285 A151286 A151287

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved