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%I #9 Jun 29 2018 22:21:22
%S 1,1,5,33,280,2883,34817,481477,7489454,129259662,2448516959,
%T 50460561330,1123192711285,26838555204646,684871918806173,
%U 18580595826856937,533846105922876855,16187892824592956798,516492582419620294678,17292646954057122160416,606075769032914504000388
%N Expansion of e.g.f. exp(exp(exp(x*exp(x)) - 1) - 1).
%C Natural numbers exponentiated thrice.
%H Alois P. Heinz, <a href="/A316158/b316158.txt">Table of n, a(n) for n = 0..423</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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%p a:= (proc(p) local g; g:= proc(n) option remember; `if`(n=0, 1,
%p p(n)+add(binomial(n-1, k-1)*p(k)*g(n-k), k=1..n-1))
%p end end@@3)(j-> j):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jun 25 2018
%t nmax = 20; CoefficientList[Series[Exp[Exp[Exp[x Exp[x]] - 1] - 1], {x, 0, nmax}], x] Range[0, nmax]!
%t b[n_] := b[n] = Sum[k^(n - k) Binomial[n, k] BellB[k], {k, n}]; a[n_] := a[n] = Sum[b[k] Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]
%Y Cf. A000248, A000258, A007550.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jun 25 2018
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