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Expansion of e.g.f. Product_{k>=1} B(x^k)^(1/k!) where B(x) = exp(exp(x) - 1) = e.g.f. of Bell numbers.
2

%I #27 Jul 07 2021 11:24:14

%S 1,1,3,9,41,183,1145,6835,52043,398441,3577291,32395905,342875813,

%T 3603992759,42817702673,518311440987,6897155535843,93092680608025,

%U 1376879589495555,20561329595474713,333009853668160757,5480574201430489831,96322698607644959065

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

%H Seiichi Manyama, <a href="/A346037/b346037.txt">Table of n, a(n) for n = 0..464</a>

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

%F E.g.f.: exp( Sum_{k>=1} A121860(k)*x^k/k! ).

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

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

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

%Y Cf. A000110, A121860, A209902, A209903, A346039, A346056.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jul 02 2021