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

0,3

Comments

Hankel transform is A165202. Essentially the same as A026674.

Links

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

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

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 *)

Prog

(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
(Magma) R<x>:=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
(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
(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

Crossrefs

Cf. A000108, A026674, A165202.

Sequence in context: A052927 A012781 A132820 * A026674 A099781 A026872

Adjacent sequences: A165198 A165199 A165200 * A165202 A165203 A165204

Keyword

easy,nonn

Author

Paul Barry, Sep 07 2009

Status

approved