A088369 Expansion of e.g.f. 1/(1 - x - x^2)^x.

1, 0, 2, 9, 44, 390, 3474, 37800, 471344, 6602904, 103271400, 1779944760, 33542915592, 686101244400, 15139184749584, 358465510133640, 9066087526045440, 243928110816129600, 6956913949298380224, 209651038286581756800, 6656701196017929467520, 222116657005058778103680

Offset: 0

Author

Paul D. Hanna, Sep 28 2003

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A191422.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 1/(1-x-x^2)^x ))); // G. C. Greubel, Dec 12 2022

Mathematica

With[{nn=30},CoefficientList[Series[1/(1-x-x^2)^x,{x,0,nn}],x]Range[ 0,nn]!] (* Harvey P. Dale, May 06 2012 *)

PARI

my(x='x+O('x^22)); Vec(serlaplace(1/(1-x-x^2)^x)) \\ Joerg Arndt, Dec 13 2022

SageMath

def A088369_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x*log(1-x-x^2)) ).egf_to_ogf().list()
A088369_list(40) # G. C. Greubel, Dec 12 2022

Formula

a(n) ~ n! * n^(c-1) / (Gamma(c) * 5^(c/2) * c^c * c^n), where c = (sqrt(5)-1)/2. - Vaclav Kotesovec, Nov 05 2014

a(n) = n! * Sum_{j=0..n} Sum_{k=0..j} binomial(j,n-j-k) * |Stirling1(j,k)|/j!. - Seiichi Manyama, Mar 13 2024

Extensions

Definition corrected by Vaclav Kotesovec, Nov 05 2014