A088369 - OEIS

%I #23 Mar 13 2024 12:13:54

%S 1,0,2,9,44,390,3474,37800,471344,6602904,103271400,1779944760,

%T 33542915592,686101244400,15139184749584,358465510133640,

%U 9066087526045440,243928110816129600,6956913949298380224,209651038286581756800,6656701196017929467520,222116657005058778103680

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

%H G. C. Greubel, <a href="/A088369/b088369.txt">Table of n, a(n) for n = 0..400</a>

%F 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

%F 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

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

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 1/(1-x-x^2)^x ))); // _G. C. Greubel_, Dec 12 2022

%o (SageMath)

%o def A088369_list(prec):

%o     P.<x> = PowerSeriesRing(QQ, prec)

%o     return P( exp(-x*log(1-x-x^2)) ).egf_to_ogf().list()

%o A088369_list(40) # _G. C. Greubel_, Dec 12 2022

%o (PARI) my(x='x+O('x^22)); Vec(serlaplace(1/(1-x-x^2)^x)) \\ _Joerg Arndt_, Dec 13 2022

%Y Cf. A191422.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 28 2003

%E Definition corrected by _Vaclav Kotesovec_, Nov 05 2014