A318181 Expansion of e.g.f. exp((1 - exp(-6*x))/6).

1, 1, -5, 19, 1, -1103, 15211, -123821, 120865, 19464193, -474727877, 7017193075, -50549088671, -931708750607, 49742453940331, -1276858353426317, 21239149342811329, -100057086073774463, -9091588769200298501, 454849803186974314579, -13529950476868715792063, 262961916344710204693681

Offset: 0

Author

Ilya Gutkovskiy, Aug 20 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..447
• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Cf. A005012, A008542, A009235, A014182, A317996, A318179, A318180.

Programs

Maple

seq(n!*coeff(series(exp((1-exp(-6*x))/6),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 21; CoefficientList[Series[Exp[(1 - Exp[-6 x])/6], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-6)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-6)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Table[(-6)^n BellB[n, -1/6], {n, 0, 21}] (* Peter Luschny, Aug 20 2018 *)

Formula

a(n) = Sum_{k=0..n} (-6)^(n-k)*Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-6)^(k-1)*binomial(n-1,k-1)*a(n-k).

a(n) = (-6)^n*BellPolynomial_n(-1/6). - Peter Luschny, Aug 20 2018