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A001189
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Number of degree-n permutations of order exactly 2.
(Formerly M2801 N1127)
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33
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0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503, 2390479, 10349535, 46206735, 211799311, 997313823, 4809701439, 23758664095, 119952692895, 618884638911, 3257843882623, 17492190577599, 95680443760575
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Number of set partitions of [n] into blocks of size 2 and 1 with at least one block of size 2. - Olivier GERARD (olivier.gerard(AT)gmail.com), Oct 29 2007
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REFERENCES
| R. B. Herrera, The number of elements of given period in finite symmetric group, Amer. Math. Monthly 64, 1957, 488-490.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Thanatipanonda, Thotsaporn, Inversions and major index for permutations, Math. Mag., No. 4, 2004
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FORMULA
| a(n) = b(n, 2), where b(n, d)=Sum_{k=1..n} (n-1)!/(n-k)! * Sum_{l:lcm{k, l}=d} b(n-k, l), b(0, 1)=1 is the number of degree-n permutations of order exactly d.
E.g.f.: -exp(x)+exp(x+1/2*x^2).
a(n) = a(n-1)+(1+a(n-2))*(n-1) = Sum_{j = 1 to floor[n/2]}[n!/(j!*(n-2j)!*(2^j))] = A000085(n)-1. - Henry Bottomley (se16(AT)btinternet.com), May 03 2001
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MATHEMATICA
| RecurrenceTable[{a[1]==0, a[2]==1, a[n]==a[n-1]+(1+a[n-2])(n-1)}, a[n], {n, 25}] (* From Harvey P. Dale, Jul 27 2011 *)
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CROSSREFS
| Equals A000085 - 1. Cf. A001470 - A001473, A052501, A053496-A053504, A061121-A061128.
Sequence in context: A183111 A132835 A191354 * A198180 A101786 A192371
Adjacent sequences: A001186 A001187 A001188 * A001190 A001191 A001192
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 14 2001
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