A151282 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, 0), (0, 1), (1, 1)}.
1, 2, 6, 18, 58, 190, 638, 2170, 7474, 25974, 90982, 320738, 1137002, 4049838, 14485326, 52001290, 187292514, 676546790, 2450311862, 8895769714, 32366225562, 117995832990, 430960312862, 1576675041434, 5777325893266, 21200338220630, 77901645076998, 286615385651970, 1055762834791114, 3893279267979662
Offset: 0
Examples
G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 58*x^4 + 190*x^5 + 638*x^6 + 2170*x^7 + ...
Links
- Robert Israel, Table of n, a(n) for n = 0..200
- 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.
- Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
- M. Bousquet-Mélou and M. Mishna, 2008. Walks with small steps in the quarter plane, ArXiv 0810.4387.
Programs
-
Maple
b:= proc(n, l) option remember; `if`(-1 in {l[]}, 0, `if`(n=0, 1, add(b(n-1, l+d), d=[[-1, -1], [-1, 0], [0, 1], [1, 1]]))) end: a:= n-> b(n, [0$2]): seq(a(n), n=0..40); # Alois P. Heinz, Feb 18 2013 -
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[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}]
Formula
Conjecture: (n+1)*a(n)-3*(2n+1)*a(n-1) +(n+10)*a(n-2) +28(n-2)*a(n-3)=0. - R. J. Mathar, Dec 08 2011
a(n) ~ sqrt(1348+953*sqrt(2)) * (1+2*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 14 2013
Comments