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Expansion of e.g.f. exp(Sum_{p prime} p * x^p).
3

%I #19 Apr 28 2022 09:50:40

%S 1,0,4,18,48,1320,4200,115920,994560,11793600,264055680,2601244800,

%T 67761429120,1067726499840,21513457405440,485310649824000,

%U 9925206939648000,254012624170905600,6174538264806912000,160933619800835481600,4458470291543671603200

%N Expansion of e.g.f. exp(Sum_{p prime} p * x^p).

%F a(0) = 1; a(n) = (n-1)! * Sum_{p<=n, p prime} p^2 * a(n-p)/(n-p)!.

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, isprime(k)*k*x^k))))

%o (PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, isprime(k)*k^2*a(n-k)/(n-k)!));

%Y Cf. A000040, A082579, A218002, A351940, A353163, A353164.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 28 2022