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A004700 Expansion of e.g.f. 1/(3 - exp(x) - exp(2*x)).

%I A004700 #31 Sep 08 2022 08:44:33
%S A004700 1,3,23,261,3947,74613,1692563,44794221,1354849547,46101247173,
%T A004700 1742977452803,72487571292381,3288697207653947,161639067567489333,
%U A004700 8555659001848069043,485203383272476257741,29350999686572204663147,1886474390059466622333093
%N A004700 Expansion of e.g.f. 1/(3 - exp(x) - exp(2*x)).
%H A004700 Vincenzo Librandi, <a href="/A004700/b004700.txt">Table of n, a(n) for n = 0..200</a>
%F A004700 G.f.: 1/(3 - E(0)), where E(k)= 1 + 2^k/(1 - x/(x + 2^k*(k+1)/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 21 2013
%F A004700 a(n) ~ 2*n!/((13-sqrt(13))*(log((sqrt(13)-1)/2))^(n+1)). - _Vaclav Kotesovec_, Aug 13 2013
%F A004700 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2^k + 1) * a(n-k). - _Ilya Gutkovskiy_, Jan 15 2020
%p A004700 seq(coeff(series(factorial(n)*(3-exp(x)-exp(2*x))^(-1),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Oct 10 2018
%t A004700 With[{nn=20},CoefficientList[Series[1/(3-Exp[x]-Exp[2x]),{x,0,nn}], x] Range[0,nn]!] (* _Harvey P. Dale_, Dec 04 2011 *)
%o A004700 (PARI) x='x+O('x^30); Vec(serlaplace(1/(3-sum(k=1,2, exp(k*x))))) \\ _G. C. Greubel_, Oct 09 2018
%o A004700 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(3-Exp(x)-Exp(2*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Oct 09 2018
%Y A004700 Column k=2 of A320253.
%K A004700 nonn
%O A004700 0,2
%A A004700 _N. J. A. Sloane_