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%I #19 Jul 15 2022 20:37:18
%S 0,1,9,52,283,1561,8930,53411,334785,2199034,15119621,108644581,
%T 814474176,6358910949,51615342685,434865155292,3796991928727,
%U 34308796490005,320379418256794,3087939032182127,30683582797977749,313977721545709002,3305220440084030809,35759627532783842561
%N E.g.f.: exp(exp(x) - 1)*(exp(3*x) - 2*exp(x) + 1).
%C Stirling transform of the cubes (A000578).
%C Exponential convolution of the sequences A000110 and A058481 (with a(0) = 0).
%H Alois P. Heinz, <a href="/A282179/b282179.txt">Table of n, a(n) for n = 0..572</a>
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F a(n) = Sum_{k=0..n} Stirling2(n,k)*A000578(k).
%F a(n) = A000110(n) + A005494(n) - A186021(n+1).
%e E.g.f.: A(x) = x/1! + 9*x^2/2! + 52*x^3/3! + 283*x^4/4! + 1561*x^5/5! + 8930*x^6/6! + ...
%p b:= proc(n, m) option remember; `if`(n=0,
%p m^3, m*b(n-1, m)+b(n-1, m+1))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..27); # _Alois P. Heinz_, Jul 15 2022
%t Range[0, 23]! CoefficientList[Series[Exp[Exp[x] - 1] (Exp[3 x] - 2 Exp[x] + 1), {x, 0, 23}], x]
%t Table[Sum[StirlingS2[n, k] k^3, {k, 0, n}], {n, 0, 23}]
%t Table[Sum[Binomial[n, k] BellB[n-k] (3^k - 2), {k, 1, n}], {n, 0, 23}]
%t Table[BellB[n+3] - 3*BellB[n+2] + BellB[n], {n, 0, 23}] (* _Vaclav Kotesovec_, Aug 06 2021 *)
%Y Cf. A000110, A000578, A005494, A033452, A058481, A186021.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Feb 08 2017
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