A302397 Expansion of e.g.f. 1/(1 + x*exp(x)).

1, -1, 0, 3, -4, -25, 114, 287, -4152, 1647, 192230, -807961, -10164804, 111209111, 454840554, -14657978385, 21202175504, 1988791958879, -15488971798194, -260886468394153, 4872247004699460, 23537372210149959, -1365745577227898350, 4274609859520565663, 364461939727273277016

Offset: 0

Author

Ilya Gutkovskiy, Apr 07 2018

Keywords

• sign

Examples

			1/(1 + x*exp(x)) = 1 - x/1! + 3*x^3/3! - 4*x^4/4! - 25*x^5/5! + 114*x^6/6! + 287*x^7/7! - 4152*x^8/8! + 1647*x^9/9! + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..474

Crossrefs

Cf. A006153, A009306, A089148.

Programs

Maple

a:=series(1/(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/(1 + x Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(-1)^(n - k) (n - k)^k/k!, {k, 0, n}], {n, 24}]]
Join[{1}, Table[Sum[(-1)^k k! k^(n - k) Binomial[n, k], {k, 0, n}], {n, 24}]]

Formula

E.g.f.: 1/(1 + x*exp(x)).

a(n) = n!*Sum_{k=0..n} (-1)^(n-k)*(n-k)^k/k!.

a(n) = Sum_{k=0..n} (-1)^k*k!*k^(n-k)*binomial(n,k).