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A351940 a(n) is the number of partitions of the set {1,2,...,n} into lists having a prime number of elements.

%I A351940 #24 Mar 03 2022 11:46:41
%S A351940 1,0,2,6,12,240,480,12600,62160,665280,10009440,94802400,1497545280,
%T A351940 23662679040,317854817280,5236146115200,102522189369600,
%U A351940 1772579589580800,39459091697625600,809304973699622400,17463916757211724800,388537548150495744000
%N A351940 a(n) is the number of partitions of the set {1,2,...,n} into lists having a prime number of elements.
%F A351940 E.g.f.: Product_{k > 0} exp(x^prime(k)).
%F A351940 E.g.f.: exp(Sum_{k > 0} x^prime(k)).
%F A351940 a(0) = 1; a(n) = Sum_{p<=n, p prime} p! * binomial(n-1,p-1) * a(n-p).
%e A351940 a(2) =   2 : (12) (2! * 1 ways).
%e A351940 a(3) =   6 : (123) (3! * 1 ways).
%e A351940 a(4) =  12 : (12)(34) (2! * 2! * 3 ways).
%e A351940 a(5) = 240 : (12345) (5! * 1 ways), (123)(45) (3! * 2! * 10 ways).
%e A351940 a(6) = 480 : (123)(456) (3! * 3! * 10 ways), (12)(34)(56) (2! * 2! * 2! * 15 ways).
%o A351940 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, isprime(k)*x^k))))
%o A351940 (PARI) a(n) = if(n==0, 1, sum(k=1, n, isprime(k)*k!*binomial(n-1, k-1)*a(n-k)));
%Y A351940 Cf. A000262, A010051, A190476.
%K A351940 nonn
%O A351940 0,3
%A A351940 _Seiichi Manyama_, Feb 26 2022