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

0,5

Comments

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

Links

Table of n, a(n) for n=0..33.

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000296, A123023.

Sequence in context: A110713 A098702 A100001 * A299754 A190529 A343584

Adjacent sequences: A227934 A227935 A227936 * A227938 A227939 A227940

Keyword

nonn,easy

Author

David Eppstein, Oct 06 2013

Status

approved