A110038 The partition function G(n,5).
1, 1, 2, 5, 15, 52, 202, 869, 4075, 20645, 112124, 648649, 3976633, 25719630, 174839120, 1245131903, 9263053753, 71806323461, 578719497070, 4839515883625, 41916097982471, 375401824277096, 3471395994487422, 33099042344383885, 325005134436155395
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..591 (terms 0..200 from Alois P. Heinz)
- 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.
- Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
- 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, 5): seq(a(n), n=0..30); # Alois P. Heinz, Apr 20 2012 # second Maple program: a:= proc(n) option remember; `if`(n<5, [1, 1, 2, 5, 15][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))))) 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, 5]; 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 ).
a(n) = n! * sum(k=1..n, 1/k! * sum(r=0..k, C(k,r) * sum(m=0..r, 2^(m-r) * C(r,m) * sum(j=0..m, C(m,j) * C(j,n-m-k-j-r) * 6^(j-m) * 24^(n-r-m-k-2*j) * 120^(m+k+j+r-n))))). - Vladimir Kruchinin, Jan 25 2011
a(n) = G(n,5) 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