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

1, 0, 2, 6, 12, 240, 480, 12600, 62160, 665280, 10009440, 94802400, 1497545280, 23662679040, 317854817280, 5236146115200, 102522189369600, 1772579589580800, 39459091697625600, 809304973699622400, 17463916757211724800, 388537548150495744000

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

E.g.f.: Product_{k > 0} exp(x^prime(k)).

E.g.f.: exp(Sum_{k > 0} x^prime(k)).

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

Example

a(2) =   2 : (12) (2! * 1 ways).
a(3) =   6 : (123) (3! * 1 ways).
a(4) =  12 : (12)(34) (2! * 2! * 3 ways).
a(5) = 240 : (12345) (5! * 1 ways), (123)(45) (3! * 2! * 10 ways).
a(6) = 480 : (123)(456) (3! * 3! * 10 ways), (12)(34)(56) (2! * 2! * 2! * 15 ways).

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, isprime(k)*x^k))))
(PARI) a(n) = if(n==0, 1, sum(k=1, n, isprime(k)*k!*binomial(n-1, k-1)*a(n-k)));

Crossrefs

Cf. A000262, A010051, A190476.

Sequence in context: A195338 A179201 A105122 * A132076 A309743 A291062

Adjacent sequences: A351937 A351938 A351939 * A351941 A351942 A351943

Keyword

nonn

Author

Seiichi Manyama, Feb 26 2022

Status

approved