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Expansion of e.g.f. exp(Sum_{p prime} x^p / (p-1)!).
2

%I #13 Apr 28 2022 09:50:49

%S 1,0,2,3,12,65,210,1477,7560,45864,338310,2176031,17657640,139280869,

%T 1150004856,10572694860,94834041120,931995595457,9384294360168,

%U 96974005210273,1066116104926500,11838081891521760,137785102884102366,1652584041236345933

%N Expansion of e.g.f. exp(Sum_{p prime} x^p / (p-1)!).

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

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

%o (PARI) a(n) = if(n==0, 1, sum(k=1, n, isprime(k)*k*binomial(n-1, k-1)*a(n-k)));

%Y Cf. A000040, A000248, A190476, A353162.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 28 2022