A165205 a(n) = C(2n-1,n) + C(2n+1,n+1) - C(0,n).

1, 4, 13, 45, 161, 588, 2178, 8151, 30745, 116688, 445094, 1704794, 6552378, 25258600, 97617060, 378098955, 1467343305, 5704370760, 22210199550, 86595896310, 338052201630, 1321178419080, 5168764845660, 20240517205350

Offset: 0

Author

Paul Barry, Sep 07 2009

Keywords

• easy
• nonn

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

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+x)*(1-Sqrt(1-4*x))/(2*x*Sqrt(1-4*x)) )); // G. C. Greubel, Jul 18 2019

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 *)

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

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

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