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A344623
Pseudo-involution companion for the Fibonacci generating function.
1
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
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
KEYWORD
nonn
AUTHOR
Alexander Burstein, May 24 2021
STATUS
approved