OFFSET
0,2
COMMENTS
From Paul Barry, Jan 19 2009: (Start)
Hankel transform is 2^C(n+1,2).
Row sums of Riordan array ((1-2x)/(1-x+2x^2),x(1-x)/(1-x+2x^2))^{-1}.
G.f.: 1/(1-2x-2x^2/(1-x-2x^2/(1-x-2x^2/(1-x-2x^2/(1-...))))) (continued fraction).
First column of Riordan array ((1-x)/(1+x+2x^2),x/(1+x+2x^2))^{-1}. (End)
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.
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
EXAMPLE
G.f. = 1 + 2*x + 6*x^2 + 18*x^3 + 58*x^4 + 190*x^5 + 638*x^6 + 2170*x^7 + ...
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}]
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Manuel Kauers, Nov 18 2008
STATUS
approved