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A347339 E.g.f.: exp( (exp(x) - 1) * exp(exp(x) - 1) ).

%I A347339 #10 Sep 02 2021 15:39:10
%S A347339 1,1,4,20,123,902,7656,73509,785154,9213324,117624569,1621028312,
%T A347339 23959376436,377730250003,6322478398476,111904530008040,
%U A347339 2087093471665987,40891426070289970,839329531471890724,18004595602417946685,402747680140030433886,9376084240910510840672,226760664399026618376569
%N A347339 E.g.f.: exp( (exp(x) - 1) * exp(exp(x) - 1) ).
%C A347339 Exponential transform of A138378.
%C A347339 Stirling transform of A000248.
%H A347339 Alois P. Heinz, <a href="/A347339/b347339.txt">Table of n, a(n) for n = 0..465</a>
%F A347339 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * (Bell(k+1) - Bell(k)) * a(n-k).
%F A347339 a(n) = Sum_{k=0..n} Stirling2(n,k) * A000248(k).
%p A347339 g:= proc(n) option remember; `if`(n=0, 1,
%p A347339       add(g(n-j)*j*binomial(n-1, j-1), j=1..n))
%p A347339     end:
%p A347339 b:= proc(n, m) option remember; `if`(n=0,
%p A347339       g(m), m*b(n-1, m)+b(n-1, m+1))
%p A347339     end:
%p A347339 a:= n-> b(n, 0):
%p A347339 seq(a(n), n=0..22);  # _Alois P. Heinz_, Aug 27 2021
%t A347339 nmax = 22; CoefficientList[Series[Exp[(Exp[x] - 1) Exp[Exp[x] - 1]], {x, 0, nmax}], x] Range[0, nmax]!
%t A347339 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (BellB[k + 1] - BellB[k]) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
%o A347339 (PARI) my(x='x+O('x^25)); Vec(serlaplace(exp((exp(x)-1)*exp(exp(x)-1)))) \\ _Michel Marcus_, Aug 27 2021
%Y A347339 Cf. A000110, A000248, A000258, A005493, A005727, A087761, A138378.
%K A347339 nonn
%O A347339 0,3
%A A347339 _Ilya Gutkovskiy_, Aug 27 2021