A165201 Expansion of 1/(1-x*c(x)^3), c(x) the g.f. of A000108.

1, 1, 4, 16, 65, 267, 1105, 4597, 19196, 80380, 337284, 1417582, 5965622, 25130844, 105954110, 447015744, 1886996681, 7969339643, 33670068133, 142301618265, 601586916703, 2543852427847, 10759094481491, 45513214057191

Offset: 0

Author

Paul Barry, Sep 07 2009

Keywords

• easy
• nonn

Comments

Hankel transform is A165202. Essentially the same as A026674.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000108, A026674, A165202.

Programs

GAP

List([0..30], n-> (1/2)*(2*0^n + Fibonacci(3*n-2) + Sum([0..n], j-> Binomial(2*j, j)*Fibonacci(3*(n-j)+1)/(1-2*j) ))); # G. C. Greubel, Jul 18 2019

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-3*x-2*x^2 + (1-x)*Sqrt(1-4*x))/(2*(1-4*x-x^2)) )); // G. C. Greubel, Jul 18 2019

Mathematica

CoefficientList[Series[(1-3*x-2*x^2+(1-x)*Sqrt[1-4*x])/(2*(1-4*x-x^2)), {x, 0, 30}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PARI

my(x='x+O('x^30)); Vec((1-3*x-2*x^2 + (1-x)*sqrt(1-4*x))/(2*(1-4*x-x^2))) \\ G. C. Greubel, Jul 18 2019

Sage

((1-3*x-2*x^2 + (1-x)*sqrt(1-4*x))/(2*(1-4*x-x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 18 2019

Formula

G.f.: (1-3*x-2*x^2 + (1-x)*sqrt(1-4*x))/(2*(1-4*x-x^2)).

a(n) = (1/2)*Sum_{k=0..n} C(2k,k)*F(3(n-k)+1)/(1-2k) + (1/2)*(F(3n-2) + 2*0^n).

Conjecture: n*(n-3)*a(n) +2*(-4*n^2+15*n-10)*a(n-1) +(15*n^2-69*n+80)*a(n-2) +2*(n-2)*(2*n-5)*a(n-3) =0. - R. J. Mathar, Nov 15 2011

a(n) ~ 1/10*(3*sqrt(5)-5)*(sqrt(5)+2)^n. - Vaclav Kotesovec, Oct 20 2012