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

Author

Alexander Burstein, May 24 2021

Keywords

• nonn

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

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

Crossrefs

Cf. A000045, A000108, A200031.

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