A006155 Expansion of e.g.f.: 1/(2-x-exp(x)).

1, 2, 9, 61, 551, 6221, 84285, 1332255, 24066691, 489100297, 11044268633, 274327080611, 7433424980943, 218208342366093, 6898241919264181, 233651576126946103, 8441657595745501019, 324052733365292875025, 13171257161208184782225, 565092918793429218839307

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..396
• S. Getu and L. W. Shapiro, Combinatorial view of the composition of functions, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 157

Crossrefs

Cf. A032112.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( 1/(2-x-Exp(x)) ))); // G. C. Greubel, Jan 09 2025

Mathematica

With[{nn=20},CoefficientList[Series[1/(2-x-E^x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 27 2018 *)

SageMath

def A006155_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1/(2-x-exp(x)) ).egf_to_ogf().list()
print(A006155_list(40)) # G. C. Greubel, Jan 09 2025

Formula

E.g.f.: 1/(2-x-exp(x)).

a(n) ~ n! / ((1+c) * (2-c)^(n+1)), where c = A226571 = LambertW(exp(2)). - Vaclav Kotesovec, Jun 06 2019

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020

Extensions

More terms from Ralf Stephan, Mar 12 2004