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 A306948 Expansion of e.g.f. (1 + x)*log(1 + x)*exp(x). 0

%I

%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

%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

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Last modified January 22 04:27 EST 2020. Contains 331133 sequences. (Running on oeis4.)