A344623 Pseudo-involution companion for the Fibonacci generating function.

1, 3, 9, 32, 126, 538, 2429, 11412, 55201, 272993, 1373784, 7011297, 36201841, 188761743, 992491049, 5256244537, 28013213196, 150128293038, 808543940999, 4373798584407, 23753913152691, 129469596050953, 707969244301884, 3882857013894482, 21353585584100401

Offset

0,2

Comments

a(n) is the number of colored Schröder paths of semilength n with steps U=(1,0) and D=(1,-1) of 1 color and H=(2,0) of 2 colors, red and blue, where H does not follow D, and no two red H steps are consecutive.

Links

Table of n, a(n) for n=0..24.

Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Formula

G.f.: A(x) satisfies A(-x*A(x)) = 1/A(x) and F(-x*A(x)) = 1/F(x), where x*F(x) is g.f. of A000045. I.e., the Riordan array (F(x), x*A(x)) is a pseudo-involution.

G.f.: A(x) = (F(x) - 1)*C(F(x) - 1)/x, where C(x) is the g.f. of A000108 and x*F(x) is g.f. of A000045.

G.f.: A(x) = (1 - sqrt((1 - 5*x - 5*x^2)/(1 - x - x^2)))/(2x).

G.f.: Let B(x) = 2 + g.f.(A200031(n)), then A(x) = 1 + x*A(x)*B(x^2*A(x)).

a(n) ~ sqrt(3) * 5^(n/2 + 1) * phi^(2*n + 1) / (8*sqrt(Pi)*n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 25 2021

Crossrefs

Cf. A000045, A000108, A200031.

Sequence in context: A104184 A339230 A193621 * A320180 A183425 A039628

Adjacent sequences: A344620 A344621 A344622 * A344624 A344625 A344626

Keyword

nonn

Author

Alexander Burstein, May 24 2021

Status

approved