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%I A110038 #39 Oct 01 2017 01:32:01
%S A110038 1,1,2,5,15,52,202,869,4075,20645,112124,648649,3976633,25719630,
%T A110038 174839120,1245131903,9263053753,71806323461,578719497070,
%U A110038 4839515883625,41916097982471,375401824277096,3471395994487422,33099042344383885,325005134436155395
%N A110038 The partition function G(n,5).
%C A110038 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 A110038 Seiichi Manyama, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0ExMTAwMzgvYjExMDAzOC50eHQ">Table of n, a(n) for n = 0..591</a> (terms 0..200 from Alois P. Heinz)
%H A110038 Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="/load/view.php?a=aHR0cDovL2FyeGl2Lm9yZy9hYnMvMTcwMS4wODM5NA">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394 [math.CO], 2017.
%H A110038 David Applegate and N. J. A. Sloane, <a href="/load/view.php?a=aHR0cDovL2FyeGl2Lm9yZy9hYnMvMDkwNy4wNTEz">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009.
%H A110038 P. J. Cameron, <a href="/load/view.php?a=aHR0cDovL3d3dy5jcy51d2F0ZXJsb28uY2Evam91cm5hbHMvSklTL1ZPTDMvZ3JvdXBzLmh0bWw">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A110038 Vladimir Kruchinin, <a href="/load/view.php?a=aHR0cDovL2FyeGl2Lm9yZy9hYnMvMTAwOS4yNTY1">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%H A110038 F. L. Miksa, L. Moser and M. Wyman, <a href="/load/view.php?a=aHR0cDovL2R4LmRvaS5vcmcvMTAuNDE1My9DTUItMTk1OC0wMTAtMg">Restricted partitions of finite sets</a>, Canad. Math. Bull., 1 (1958), 87-96.
%F A110038 E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ).
%F A110038 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 A110038 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 A110038 G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A110038        add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
%p A110038     end:
%p A110038 a:= n-> G(n, 5):
%p A110038 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 20 2012
%p A110038 # second Maple program:
%p A110038 a:= proc(n) option remember; `if`(n<5, [1, 1, 2, 5, 15][n+1],
%p A110038       a(n-1)+(n-1)*(a(n-2)+(n-2)/2*(a(n-3)+(n-3)/3*(a(n-4)
%p A110038       +(n-4)/4*a(n-5)))))
%p A110038     end:
%p A110038 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 15 2013
%t A110038 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 A110038 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 A110038 Column k=5 of A229223.
%Y A110038 Cf. A276925.
%K A110038 nonn
%O A110038 0,3
%A A110038 _N. J. A. Sloane_, May 13 2009

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