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%I #27 Jun 28 2017 13:01:47
%S 1,1,2,5,15,52,203,876,4131,21065,115274,672673,4163743,27216840,
%T 187160429,1349511178,10173555345,79982663997,654277037674,
%U 5557624876513,48931106059451,445790174654588,4196351007814659,40757862664061104,407944375184911787
%N The partition function G(n,6).
%C 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]
%H Alois P. Heinz, <a href="/A148092/b148092.txt">Table of n, a(n) for n = 0..500</a>
%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 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 + x^6/720 ).
%F 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
%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, 6):
%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<6, [1, 1, 2, 5, 15, 52][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) +(n-5)/5*a(n-6))))))
%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, 6]; 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=6 of A229223.
%Y Cf. A276926.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, May 13 2009