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Expansion of e.g.f. exp(Sum_{k>=1} prime(k)*x^k).
1

%I #41 Dec 28 2022 05:42:49

%S 1,2,10,74,676,7592,97024,1416200,23015248,412777952,8090869984,

%T 171435904928,3908548404160,95264270043776,2470715015425024,

%U 67913132377486208,1971038886452490496,60212661838223997440,1930529043247940342272,64801071784954698480128

%N Expansion of e.g.f. exp(Sum_{k>=1} prime(k)*x^k).

%F a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A033286(k) * a(n-k)/(n-k)!.

%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*

%p ithprime(j)*j*binomial(n, j)*j!, j=1..n)/n)

%p end:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Apr 28 2022

%t a[0] = 1; a[n_] := a[n] = (n-1)! Sum[k Prime[k] a[n-k]/(n-k)!, {k, 1, n}];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 28 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, prime(k)*x^k))))

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

%Y Cf. A000040, A007446, A033286, A353162, A353166.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 28 2022