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A292916 a(n) = n! * [x^n] exp(n*x)/(2 - exp(x)).

%I A292916 #17 Jun 12 2024 20:44:09
%S A292916 1,2,11,94,1083,15666,272451,5532206,128409707,3352959850,97259891163,
%T A292916 3102552150006,107936130271899,4066743353318114,164961642651034547,
%U A292916 7167348523420169278,332081754670735087275,16343667009638859878298,851478575825591156040843,46814697307371602567813126
%N A292916 a(n) = n! * [x^n] exp(n*x)/(2 - exp(x)).
%C A292916 The n-th term of the n-th binomial transform of A000670.
%H A292916 G. C. Greubel, <a href="/A292916/b292916.txt">Table of n, a(n) for n = 0..380</a>
%H A292916 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A292916 a(n) = A292915(n,n).
%F A292916 a(n) ~ n! * 2^(n-1) / (log(2))^(n+1). - _Vaclav Kotesovec_, Sep 27 2017
%F A292916 a(n) = 2^n*A000670(n) - Sum_{k=0..n-1} 2^k*(n-1-k)^n. - _Seiichi Manyama_, Dec 25 2023
%p A292916 b:= proc(n, k) option remember; k^n +add(
%p A292916        binomial(n, j)*b(j, k), j=0..n-1)
%p A292916     end:
%p A292916 a:= n-> b(n$2):
%p A292916 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 27 2017
%t A292916 Table[n! SeriesCoefficient[Exp[n x]/(2 - Exp[x]), {x, 0, n}], {n, 0, 19}]
%t A292916 Table[HurwitzLerchPhi[1/2, -n, n]/2, {n, 0, 19}]
%o A292916 (PARI) a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
%o A292916 a(n) = 2^n*a000670(n)-sum(k=0, n-1, 2^k*(n-1-k)^n); \\ _Seiichi Manyama_, Dec 25 2023
%o A292916 (Magma)
%o A292916 R<x>:=PowerSeriesRing(Rationals(), 50);
%o A292916 A292916:= func< n | Coefficient(R!(Laplace( Exp(n*x)/(2-Exp(x)) )), n) >;
%o A292916 [A292916(n): n in [0..30]]; // _G. C. Greubel_, Jun 12 2024
%o A292916 (SageMath) [factorial(n)*( exp(n*x)/(2-exp(x)) ).series(x,n+1).list()[n] for n in (0..30)] # _G. C. Greubel_, Jun 12 2024
%Y A292916 Main diagonal of A292915.
%Y A292916 Cf. A000670, A330603.
%K A292916 nonn
%O A292916 0,2
%A A292916 _Ilya Gutkovskiy_, Sep 26 2017