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

0, 1, 3, 5, 8, 9, 19, -15, 216, -1407, 11803, -108483, 1106192, -12363703, 150381243, -1977666743, 27965386320, -423158076351, 6822782712723, -116781368777867, 2114916140765496, -40404117909336247, 812091479233464131, -17130720178674680031, 378423227774537955688

Offset: 0

Author

Ilya Gutkovskiy, Mar 17 2019

Keywords

• sign

Crossrefs

Cf. A000110, A002741, A048994, A070071, A216313.

Programs

Maple

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

Mathematica

nmax = 24; CoefficientList[Series[(1 + x) Log[1 + x] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] BellB[k] k, {k, 0, n}], {n, 0, 24}]
Table[Sum[(-1)^(k - 1) Binomial[n, k] (n - k + 1) (k - 1)!, {k, 1, n}], {n, 0, 24}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(k)*k.

a(n) = Sum_{k=1..n} (-1)^(k-1)*binomial(n,k)*(n - k + 1)*(k - 1)!.

a(n) ~ exp(-1) * (-1)^n * n! / n^2. - Vaclav Kotesovec, Mar 18 2019

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