OFFSET
0,2
COMMENTS
Hankel transform is A165206.
This sequence has the following conjectural description using cluster algebras. Consider the cyclic directed graph with n vertices as a cluster seed S of type D_n. Let S_n be the set of seeds that can be obtained from S by iterated mutation of the initial cluster variables in S. Mutations of other cluster variables are not allowed. Then the cardinality of S_n is apparently A165205(n). This can be checked from n=2 (4) to n=8 (8151). - F. Chapoton, Apr 03 2021
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (1+x)*C(x)^2/(1-x*C(x)^2) = (1+x)*(1 - sqrt(1-4*x))/(2*x* sqrt(1-4*x)), where C(x) is the g.f. of A000108.
Apparently (n+1)*(5*n-2) -2*(5*n+3)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 25 2012
MATHEMATICA
CoefficientList[Series[(1+x)*(1-Sqrt[1-4*x])/(2*x*Sqrt[1-4*x]), {x, 0, 30}], x] (* G. C. Greubel, Jul 18 2019 *)
Table[Binomial[2n-1, n]+Binomial[2n+1, n+1]-Binomial[0, n], {n, 0, 30}] (* Harvey P. Dale, Apr 28 2022 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1+x)*(1-sqrt(1-4*x))/(2*x*sqrt(1-4*x))) \\ G. C. Greubel, Jul 18 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+x)*(1-Sqrt(1-4*x))/(2*x*Sqrt(1-4*x)) )); // G. C. Greubel, Jul 18 2019
(Sage) a=((1+x)*(1-sqrt(1-4*x))/(2*x*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False); [1]+a[1:] # G. C. Greubel, Jul 18 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 07 2009
STATUS
approved