A317996 Expansion of e.g.f. exp((1 - exp(-3*x))/3).

1, 1, -2, 1, 19, -128, 379, 1549, -32600, 261631, -845909, -10713602, 237695149, -2513395259, 11792378662, 151915180429, -4826456213273, 70741388773960, -558513179369297, -2833805536521839, 200720356696607416, -4256279445015662093, 54120395442382043743, -173423789950999240226

Offset: 0

Author

Ilya Gutkovskiy, Aug 20 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..495
• Eric Weisstein's World of Mathematics, Bell Polynomial

Crossrefs

Column k=3 of A309386.

Cf. A004212, A007559, A009235, A014182, A318179, A318180, A318181.

Programs

Maple

a:=series(exp((1 - exp(-3*x))/3), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 23; CoefficientList[Series[Exp[(1 - Exp[-3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-3)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = Sum[(-3)^(k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
Table[(-3)^n BellB[n, -1/3], {n, 0, 23}] (* Peter Luschny, Aug 20 2018 *)

Formula

a(n) = Sum_{k=0..n} (-3)^(n-k)*Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-3)^(k-1)*binomial(n-1,k-1)*a(n-k).

a(n) = (-3)^n BellPolynomial_n(-1/3). - Peter Luschny, Aug 20 2018