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A309084 a(n) = exp(3) * Sum_{k>=0} (-3)^k*k^n/k!.

%I A309084 #15 Sep 08 2022 08:46:21
%S A309084 1,-3,6,-3,-21,24,195,-111,-3072,-4053,57003,277854,-697539,-12261567,
%T A309084 -29861778,371727465,3511027599,2028432480,-188521156857,
%U A309084 -1470389129931,1655487186864,121873222577823,915525253963023,-2095901567014530,-103715912230195863,-836215492271268459
%N A309084 a(n) = exp(3) * Sum_{k>=0} (-3)^k*k^n/k!.
%H A309084 Seiichi Manyama, <a href="/A309084/b309084.txt">Table of n, a(n) for n = 0..571</a>
%F A309084 G.f.: Sum_{j>=0} (-3)^j*x^j / Product_{k=1..j} (1 - k*x).
%F A309084 E.g.f.: exp(3*(1 - exp(x))).
%F A309084 a(n) = Sum_{k=0..n} (-3)^k * Stirling2(n,k).
%p A309084 b:= proc(n, m) option remember; `if`(n=0,
%p A309084       (-3)^m, m*b(n-1, m)+b(n-1, m+1))
%p A309084     end:
%p A309084 a:= n-> b(n, 0):
%p A309084 seq(a(n), n=0..27);  # _Alois P. Heinz_, Jul 17 2022
%t A309084 Table[Exp[3] Sum[(-3)^k k^n/k!, {k, 0, Infinity}], {n, 0, 25}]
%t A309084 Table[BellB[n, -3], {n, 0, 25}]
%t A309084 nmax = 25; CoefficientList[Series[Sum[(-3)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
%t A309084 nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!
%o A309084 (Magma) [1] cat [(&+[((-3)^k*StirlingSecond(m, k)):k in [0..m]]):m in [1..25]]; // _Marius A. Burtea_, Jul 27 2019
%Y A309084 Column k = 3 of A292861.
%Y A309084 Cf. A000587, A027710, A213170, A309085, A317996.
%K A309084 sign
%O A309084 0,2
%A A309084 _Ilya Gutkovskiy_, Jul 11 2019