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

1, 3, 23, 261, 3947, 74613, 1692563, 44794221, 1354849547, 46101247173, 1742977452803, 72487571292381, 3288697207653947, 161639067567489333, 8555659001848069043, 485203383272476257741, 29350999686572204663147, 1886474390059466622333093

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Column k=2 of A320253.

Programs

Magma

m:=30; R:=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

Maple

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

Mathematica

With[{nn=20},CoefficientList[Series[1/(3-Exp[x]-Exp[2x]),{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Dec 04 2011 *)

PARI

x='x+O('x^30); Vec(serlaplace(1/(3-sum(k=1,2, exp(k*x))))) \\ G. C. Greubel, Oct 09 2018

Formula

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

a(n) ~ 2*n!/((13-sqrt(13))*(log((sqrt(13)-1)/2))^(n+1)). - Vaclav Kotesovec, Aug 13 2013

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2^k + 1) * a(n-k). - Ilya Gutkovskiy, Jan 15 2020