A227937 Partitions of n labeled elements into subsets of two or three elements.

1, 0, 1, 1, 3, 10, 25, 105, 385, 1540, 7245, 32725, 164395, 870870, 4689685, 27152125, 161786625, 997196200, 6443061625, 42702885225, 292938721075, 2078239413250, 15119319039825, 113390111659825, 873538909468225, 6894294734827500, 55855506234653125, 463151808682688125, 3927996120260086875, 34081631999814148750, 301951521812713898125, 2731127272307562253125, 25208456056107710010625, 237164027532948085570000

Offset: 0

Author

David Eppstein, Oct 06 2013

Keywords

• nonn
• easy

Comments

Periodic modulo two and modulo three, but appears to eventually be divisible by other prime powers.

Examples

			The five elements a, b, c, d, e have ten partitions into sets of size two or three: ab/cde, ac/bde, ad/bce, ae/bcd, bc/ade, bd/ace, be/acd, cd/abe, ce/abd, and de/abc.
		

Crossrefs

Cf. A000296, A123023.

Programs

Mathematica

Flatten[{1,RecurrenceTable[{2*a[n] - 2*(n-1)*a[n-2]-(n-2)*(n-1)*a[n-3] == 0,a[1]==0,a[2]==1,a[3]==1}, a, {n, 1, 20}]}] (* Vaclav Kotesovec, Oct 09 2013 *)

PARI

x='x+O('x^66); Vec( serlaplace( exp( x^2/2 + x^3/6 ) ) ) \\ Joerg Arndt, Oct 07 2013

Formula

a(n) = (n-1)*a(n-2) + (n-1)*(n-2)*a(n-3)/2.

E.g.f.: exp( x^2/2 + x^3/6 ). [Joerg Arndt, Oct 07 2013]

a(n) ~ n^(2*n/3) * 2^(-n/3) * exp(2/9 - 2*n/3 - (2*n)^(1/3)/3 + (2*n)^(2/3)/2)/sqrt(3) * (1 + 34/(162*(2*n)^(1/3)) - 34802/(131220*(2*n)^(2/3))). - Vaclav Kotesovec, Oct 09 2013