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

%I A006155 M1945 #34 Jun 23 2025 18:34:02
%S A006155 1,2,9,61,551,6221,84285,1332255,24066691,489100297,11044268633,
%T A006155 274327080611,7433424980943,218208342366093,6898241919264181,
%U A006155 233651576126946103,8441657595745501019,324052733365292875025,13171257161208184782225,565092918793429218839307
%N A006155 Expansion of e.g.f.: 1/(2-x-exp(x)).
%D A006155 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006155 Seiichi Manyama, <a href="/A006155/b006155.txt">Table of n, a(n) for n = 0..396</a>
%H A006155 S. Getu and L. W. Shapiro, <a href="/A006152/a006152.pdf">Combinatorial view of the composition of functions</a>, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
%H A006155 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=157">Encyclopedia of Combinatorial Structures 157</a>
%F A006155 E.g.f.: 1/(2-x-exp(x)).
%F A006155 a(n) ~ n! / ((1+c) * (2-c)^(n+1)), where c = A226571 = LambertW(exp(2)). - _Vaclav Kotesovec_, Jun 06 2019
%F A006155 a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k). - _Ilya Gutkovskiy_, Jul 02 2020
%t A006155 With[{nn=20},CoefficientList[Series[1/(2-x-E^x),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Apr 27 2018 *)
%o A006155 (Magma)
%o A006155 R<x>:=PowerSeriesRing(Rationals(), 40);
%o A006155 Coefficients(R!(Laplace( 1/(2-x-Exp(x)) ))); // _G. C. Greubel_, Jan 09 2025
%o A006155 (SageMath)
%o A006155 def A006155_list(prec):
%o A006155     P.<x> = PowerSeriesRing(QQ, prec)
%o A006155     return P( 1/(2-x-exp(x)) ).egf_to_ogf().list()
%o A006155 print(A006155_list(40)) # _G. C. Greubel_, Jan 09 2025
%Y A006155 Cf. A032112.
%K A006155 nonn,easy
%O A006155 0,2
%A A006155 _Simon Plouffe_
%E A006155 More terms from _Ralf Stephan_, Mar 12 2004