A151306 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), (1, 0), (1, 1)}.

1, 2, 9, 29, 139, 511, 2498, 9781, 48238, 196055, 971491, 4047652, 20113296, 85304735, 424646194, 1825369912, 9097470592, 39520352083, 197127302029, 863661309954, 4310448703844, 19018425833689, 94959280164706, 421465934703502, 2105044748278047, 9390456216012559, 46912406079449786, 210193393267663019

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..500

Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.

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

Marni Mishna and Juan Pulido, On the small-step quarter plane lattice walks with a non D-finite univariate generating function, arXiv:2605.16688 [math.CO], 2026. See p. 12 (Table 5).

Maple

steps:= [[-1, -1], [-1, 1], [-1, 0], [1, 0], [1, 1]]:
f:= proc(n, p) option remember; local t;
  if n <= min(p) then return 5^n fi;
  add(procname(n-1, t), t=remove(has, map(`+`, steps, p), -1));
end proc:
map(f, [$0..100], [0, 0]);

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, j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, 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: A268568 A150902 A150903 * A150904 A122675 A042357

Adjacent sequences: A151303 A151304 A151305 * A151307 A151308 A151309

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved