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A006505
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Number of partitions of an n-set into boxes of size >2.
(Formerly M4789)
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5
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1, 0, 0, 1, 1, 1, 11, 36, 92, 491, 2557, 11353, 60105, 362506, 2169246, 13580815, 91927435, 650078097, 4762023647, 36508923530, 292117087090, 2424048335917, 20847410586719, 185754044235873, 1711253808769653, 16272637428430152
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OFFSET
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0,7
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REFERENCES
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E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=0..25.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 102
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565
J. Riordan, Cached copy of paper
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FORMULA
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E.g.f.: exp ( exp x - 1 - x - (1/2)*x^2 ).
a(n) = sum_{k=1..[n/3]} A059022(n,k), n>=3. [From R. J. Mathar, Nov 08 2008]
a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i)/ (i+k)!))/m!); a(0)=1. [From Vladimir Kruchinin, Feb 01 2011]
Define polynomials g_n by g_0=1, g_1=g_2=0, g_3=g_4=g_5=x; g(n) = x*Sum_{i=0..n-3} binomial(n-1,i)*g_i; then a(n) = g_n(1). [Riordan]
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MAPLE
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Copy ZL := [ B, {B=Set(Set(Z, card>=3))}, labeled ]: [seq(combstruct[count](ZL, size=n), n=0..25)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 13 2007
G:={P=Set(Set(Atom, card>=3))}:combstruct[gfsolve](G, unlabeled, x):seq(combstruct[count]([P, G, labeled], size=i), i=0..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2007
g:=proc(n) option remember; if n=0 then RETURN(1); fi; if n<=2 then RETURN(0); fi; if n<=5 then RETURN(x); fi; expand(x*add(binomial(n-1, i)*g(i), i=0..n-3)); end; [seq(subs(x=1, g(n)), n=0..60)]; [N. J. A. Sloane, Jul 20 2011]
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ Exp @ x - 1 - x - x^2 / 2], {x, 0, n}]] (* Michael Somos Jul 20 2011 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) - 1 - x - x^2 / 2), n))} /* Michael Somos Jul 20 2011 */
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CROSSREFS
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Cf. A000110, A000296, A057814, A057837.
Sequence in context: A160483 A034309 A005000 * A004637 A191296 A052526
Adjacent sequences: A006502 A006503 A006504 * A006506 A006507 A006508
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Christian G. Bower, Nov 09 2000
Edited by N. J. A. Sloane, Jul 20 2011
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STATUS
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approved
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