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
Keywords
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.
Crossrefs
Programs
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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 *)
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
Comments