A165203 Expansion of (1+x)*c(x)^3/(1-x*c(x)^3), c(x) the g.f. of A000108.

1, 5, 20, 81, 332, 1372, 5702, 23793, 99576, 417664, 1754866, 7383204, 31096466, 131084954, 552969854, 2334012425, 9856336324, 41639407776, 175971686398, 743888534968, 3145439344550, 13302946909338, 56272308538682

Offset

0,2

Comments

Hankel transform is A165204.

Links

Iain Fox, Table of n, a(n) for n = 0..1595 (first 201 terms from Vincenzo Librandi)

Formula

G.f. (for offset 1): (1+x)*((1-x)*sqrt(1-4*x)+5*x-1)/(2*(1-4*x-x^2)).

a(n) = (A165201(n) - 0^n) + A165201(n+1).

Conjecture: (n+1)*(5*n-31)*a(n) +(5*n^2+74*n+62)*a(n-1) +(-285*n^2+ 1072*n-757)*a(n-2) +(695*n^2-3674*n+4206)*a(n-3) +2*(45*n-74)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Dec 11 2011

a(n) ~ (18/sqrt(5)-8) * (2+sqrt(5))^(n+2). - Vaclav Kotesovec, Feb 01 2014

Mathematica

Rest[CoefficientList[Series[(1+x)*((1-x)*Sqrt[1-4*x]+5*x-1)/(2*(1-4*x-x^2)), {x, 0, 30}], x]] (* Vaclav Kotesovec, Feb 01 2014 *)

Prog

(PARI) first(n) = x='x+O('x^(n+1)); Vec((1+x)*((1-x)*sqrt(1-4*x)+5*x-1)/(2*(1-4*x-x^2))) \\ Iain Fox, Feb 27 2018
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+x)*(1-Sqrt(1-4*x))^3/(x*(8*x^2 - (1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 18 2019
(Sage) ((1+x)*(1-sqrt(1-4*x))^3/(x*(8*x^2 - (1-sqrt(1-4*x))^3)) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 18 2019

Crossrefs

Sequence in context: A033131 A321703 A022021 * A249946 A030520 A183933

Adjacent sequences: A165200 A165201 A165202 * A165204 A165205 A165206

Keyword

easy,nonn

Author

Paul Barry, Sep 07 2009

Status

approved