A148092 The partition function G(n,6).

1, 1, 2, 5, 15, 52, 203, 876, 4131, 21065, 115274, 672673, 4163743, 27216840, 187160429, 1349511178, 10173555345, 79982663997, 654277037674, 5557624876513, 48931106059451, 445790174654588, 4196351007814659, 40757862664061104, 407944375184911787

Offset

0,3

Comments

Set partitions into sets of size at most 6. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.

David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

Formula

E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 + x^5/120 + x^6/720 ).

a(n) = G(n,6) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012

Maple

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 6):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 20 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<6, [1, 1, 2, 5, 15, 52][n+1],
      a(n-1)+(n-1)*(a(n-2) +(n-2)/2*(a(n-3) +(n-3)/3*(a(n-4)
            +(n-4)/4*(a(n-5) +(n-5)/5*a(n-6))))))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2013

Mathematica

G[n_, i_] := G[n, i] = If[n == 0, 1, If[i<1, 0, Sum[G[n-i*j, i-1] *n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; a[n_] := G[n, 6]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

Crossrefs

The sequences G(n,1), G(n,2), G(n,3), G(n,4), G(n,5), G(n,6) are given by A000012, A000085, A001680, A001681, A110038, A148092 respectively.

Column k=6 of A229223.

Cf. A276926.

Sequence in context: A366774 A287667 A192855 * A343667 A276723 A287585

Adjacent sequences: A148089 A148090 A148091 * A148093 A148094 A148095

Keyword

nonn

Author

N. J. A. Sloane, May 13 2009

Status

approved