A318180 Expansion of e.g.f. exp((1 - exp(-5*x))/5).

1, 1, -4, 11, 21, -674, 6551, -33479, -174114, 7478121, -117699599, 1090997976, 865365421, -302755297739, 7922094623596, -127940743443649, 974028543402401, 21377262410290446, -1179125036786673989, 31760741865879345821, -552216474702144564074, 2814873629049018241701

Offset: 0

Author

Ilya Gutkovskiy, Aug 20 2018

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..459
• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Cf. A005011, A008548, A009235, A014182, A317996, A318179, A318181.

Programs

Maple

seq((-5)^n*BellB(n,-1/5),n=0..30); # Robert Israel, Nov 11 2020

Mathematica

nmax = 21; CoefficientList[Series[Exp[(1 - Exp[-5 x])/5], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-5)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-5)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Table[(-5)^n BellB[n, -1/5], {n, 0, 21}] (* Peter Luschny, Aug 20 2018 *)

PARI

my(x = 'x + O('x^25)); Vec(serlaplace(exp((1 - exp(-5*x))/5))) \\ Michel Marcus, Nov 11 2020

Formula

a(n) = Sum_{k=0..n} (-5)^(n-k)*Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-5)^(k-1)*binomial(n-1,k-1)*a(n-k).

a(n) = (-5)^n*BellPolynomial_n(-1/5). - Peter Luschny, Aug 20 2018