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

%I A317996 #31 Feb 16 2025 08:33:56
%S A317996 1,1,-2,1,19,-128,379,1549,-32600,261631,-845909,-10713602,237695149,
%T A317996 -2513395259,11792378662,151915180429,-4826456213273,70741388773960,
%U A317996 -558513179369297,-2833805536521839,200720356696607416,-4256279445015662093,54120395442382043743,-173423789950999240226
%N A317996 Expansion of e.g.f. exp((1 - exp(-3*x))/3).
%H A317996 Seiichi Manyama, <a href="/A317996/b317996.txt">Table of n, a(n) for n = 0..495</a>
%H A317996 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>
%F A317996 a(n) = Sum_{k=0..n} (-3)^(n-k)*Stirling2(n,k).
%F A317996 a(0) = 1; a(n) = Sum_{k=1..n} (-3)^(k-1)*binomial(n-1,k-1)*a(n-k).
%F A317996 a(n) = (-3)^n BellPolynomial_n(-1/3). - _Peter Luschny_, Aug 20 2018
%p A317996 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 A317996 nmax = 23; CoefficientList[Series[Exp[(1 - Exp[-3 x])/3], {x, 0, nmax}], x] Range[0, nmax]!
%t A317996 Table[Sum[(-3)^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
%t A317996 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 A317996 Table[(-3)^n BellB[n, -1/3], {n, 0, 23}] (* _Peter Luschny_, Aug 20 2018 *)
%Y A317996 Column k=3 of A309386.
%Y A317996 Cf. A004212, A007559, A009235, A014182, A318179, A318180, A318181.
%K A317996 sign
%O A317996 0,3
%A A317996 _Ilya Gutkovskiy_, Aug 20 2018