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A098742
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Number of indecomposable set partitions of [1..n] without singletons.
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3
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0, 0, 1, 1, 3, 9, 33, 135, 609, 2985, 15747, 88761, 531561, 3366567, 22462017, 157363329, 1154257683, 8841865833, 70573741857, 585753925047, 5046128460801, 45044554041897, 416005748766771, 3969321053484921, 39077616720410409
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| After a(3) = 1, always divisible by 3 (in A008585). a(n) is 3 times a prime (A001748) when n = 5, 6, 11, 14, 15, 16, 19. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 22 2008
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REFERENCES
| D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.7, Problem 26.
George Puttenham, The Arte of English Poesie (1589), page 72, can be said to have stated the problem; but he omitted one case for n=5 and 22 cases for n=6, so he must have had other constraints in mind!
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..200
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FORMULA
| If f(z) is the generating function for A074664, then a(z)=zf(z)/(1+z-f(z)). Also, if g(z) is the generating function for A000296, then a(z) = 1-1/g(z).
O.g.f.: A(x) = x^2/(1-x-2*x^2/(1-2*x-3*x^2/(1-3*x-4*x^2/(1-... -n*x-(n+1)*x^2/(1- ...)))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 17 2006
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EXAMPLE
| a(5)=9 because of the set partitions 135|24, 134|25, 125|34, 145|23, 15|234, 13|245, 124|35, 12345, 14|235. [Puttenham missed the last of these.]
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MAPLE
| F:= proc(n) option remember; convert (series (1-1/add (coeff (series (exp (exp(x)-1), x, n+1), x, j)*j!*x^j, j=0..n), x, n+1), polynom) end: a:= n-> coeff (series (x*F(n)/(1+x-F(n)), x, n+1), x, n): seq (a(n), n=0..24); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 05 2008]
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MATHEMATICA
| f[n_] := f[n] = Normal[ Series[ 1-1/Sum[ SeriesCoefficient[ Series[ Exp[Exp[x] - 1], {x, 0, n + 1}], {x, 0, j}]*j!*x^j, {j, 0, n}], {x, 0, n + 1}]]; a[0] = 0; a[n_] := SeriesCoefficient[ Series[ x*(f[n]/(1 + x - f[n])), {x, 0, n + 1}], {x, 0, n}]; Table[a[n], {n, 0, 24}] (* From Jean-François Alcover, translated from Maple *)
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CROSSREFS
| Cf. A000296, A074664.
Cf. A001748, A008585.
Sequence in context: A084508 A151043 A151044 * A009212 A153344 A193110
Adjacent sequences: A098739 A098740 A098741 * A098743 A098744 A098745
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KEYWORD
| nice,nonn,easy
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AUTHOR
| D. E. Knuth, Oct 01 2004
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 21 2004
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