|
%I #21 Jul 09 2021 08:08:05
%S 1,1,3,10,47,246,1617,11586,97463,891610,9189623,102024396,1250714445,
%T 16351489116,232261545869,3499469551402,56582677946675,
%U 964734301550142,17509882651329087,333381717125596692,6710286637806825557,141167551783524139468
%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="/A346055/b346055.txt">Table of n, a(n) for n = 0..449</a>
%F E.g.f.: exp( Sum_{k>=1} (exp(x^k/k) - 1) ).
%F E.g.f.: exp( Sum_{k>=1} A005225(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, A005225, A081066, A209903, A346056, A346057.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jul 02 2021
|
