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A110038 The partition function G(n,5). 8
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 (list; graph; refs; listen; history; text; internal format)
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]

REFERENCES

F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.

David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 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

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-Fran├žois Alcover, Mar 17 2014, after Alois P. Heinz *)

CROSSREFS

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.

Column k=5 of A229223.

Cf. A276925.

Sequence in context: A220913 A287666 A158829 * A276722 A287584 A287277

Adjacent sequences:  A110035 A110036 A110037 * A110039 A110040 A110041

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, May 13 2009

STATUS

approved

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Last modified June 23 13:14 EDT 2017. Contains 288665 sequences.