A110038 - OEIS

%I #39 Oct 01 2017 01:32:01

%S 1,1,2,5,15,52,202,869,4075,20645,112124,648649,3976633,25719630,

%T 174839120,1245131903,9263053753,71806323461,578719497070,

%U 4839515883625,41916097982471,375401824277096,3471395994487422,33099042344383885,325005134436155395

%N The partition function G(n,5).

%C 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]

%H Seiichi Manyama, <a href="/A110038/b110038.txt">Table of n, a(n) for n = 0..591</a> (terms 0..200 from Alois P. Heinz)

%H Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1701.08394">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394 [math.CO], 2017.

%H David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009.

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%H F. L. Miksa, L. Moser and M. Wyman, <a href="http://dx.doi.org/10.4153/CMB-1958-010-2">Restricted partitions of finite sets</a>, Canad. Math. Bull., 1 (1958), 87-96.

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

%F 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

%F 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

%p G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p        add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))

%p     end:

%p a:= n-> G(n, 5):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 20 2012

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<5, [1, 1, 2, 5, 15][n+1],

%p       a(n-1)+(n-1)*(a(n-2)+(n-2)/2*(a(n-3)+(n-3)/3*(a(n-4)

%p       +(n-4)/4*a(n-5)))))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 15 2013

%t 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_ *)

%Y 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.

%Y Column k=5 of A229223.

%Y Cf. A276925.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, May 13 2009