OFFSET
0,3
COMMENTS
Set partitions into sets of size at most 5. 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
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.
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
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, May 13 2009
STATUS
approved