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Expansion of e.g.f. Product_{k>=1} B(x^k/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.
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%I #20 Jul 08 2021 10:59:57

%S 1,1,3,9,38,168,915,5225,34228,236622,1805297,14498751,125907798,

%T 1146476984,11129874215,112934907867,1209762361679,13499714095281,

%U 157931096158594,1918777335806274,24309294470496502,318987321135326838,4346474397776153974

%N Expansion of e.g.f. Product_{k>=1} B(x^k/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.

%H Seiichi Manyama, <a href="/A346056/b346056.txt">Table of n, a(n) for n = 0..520</a>

%F E.g.f.: exp( Sum_{k>=1} (exp(x^k/k!) - 1) ).

%F E.g.f.: exp( Sum_{k>=1} A038041(k)*x^k/k! ).

%F a(n) = (n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)!^d)) * a(n-k)/(n-k)! for n > 0.

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k/k!)-1))))

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, exp(x^k/k!)-1))))

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)!^d))*x^k))))

%o (PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!^d))*a(n-k)/(n-k)!));

%Y Cf. A000110, A038041, A081066, A209903, A346055, A346058.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jul 02 2021