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%I #13 Oct 29 2021 07:39:58
%S 0,1,3,5,8,9,19,-15,216,-1407,11803,-108483,1106192,-12363703,
%T 150381243,-1977666743,27965386320,-423158076351,6822782712723,
%U -116781368777867,2114916140765496,-40404117909336247,812091479233464131,-17130720178674680031,378423227774537955688
%N Expansion of e.g.f. (1 + x)*log(1 + x)*exp(x).
%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(k)*k.
%F a(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(n,k)*(n - k + 1)*(k - 1)!.
%F a(n) ~ exp(-1) * (-1)^n * n! / n^2. - _Vaclav Kotesovec_, Mar 18 2019
%F Conjecture: D-finite with recurrence a(n) +(n-5)*a(n-1) +(-3*n+10)*a(n-2) +3*(n-3)*a(n-3) +(-n+3)*a(n-4)=0. - _R. J. Mathar_, Aug 20 2021
%p a:=series((1 + x)*log(1 + x)*exp(x),x=0,25): seq(n!*coeff(a, x, n), n=0..24); # _Paolo P. Lava_, Mar 26 2019
%t nmax = 24; CoefficientList[Series[(1 + x) Log[1 + x] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[StirlingS1[n, k] BellB[k] k, {k, 0, n}], {n, 0, 24}]
%t Table[Sum[(-1)^(k - 1) Binomial[n, k] (n - k + 1) (k - 1)!, {k, 1, n}], {n, 0, 24}]
%Y Cf. A000110, A002741, A048994, A070071, A216313.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Mar 17 2019