%I #30 Jul 28 2019 16:53:45

%S 1,1,-2,1,19,-128,379,1549,-32600,261631,-845909,-10713602,237695149,

%T -2513395259,11792378662,151915180429,-4826456213273,70741388773960,

%U -558513179369297,-2833805536521839,200720356696607416,-4256279445015662093,54120395442382043743,-173423789950999240226

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

%H Seiichi Manyama, <a href="/A317996/b317996.txt">Table of n, a(n) for n = 0..495</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>

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

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

%F a(n) = (-3)^n BellPolynomial_n(-1/3). - _Peter Luschny_, Aug 20 2018

%p 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

%t nmax = 23; CoefficientList[Series[Exp[(1 - Exp[-3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[(-3)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]

%t 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}]

%t Table[(-3)^n BellB[n, -1/3], {n, 0, 23}] (* _Peter Luschny_, Aug 20 2018 *)

%Y Column k=3 of A309386.

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

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Aug 20 2018