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A088994
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Number of permutations in the symmetric group S_n such that the size of their centralizer is odd.
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14
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1, 1, 0, 2, 8, 24, 144, 720, 8448, 64512, 576000, 5529600, 74972160, 887546880, 11285084160, 168318259200, 2843121254400, 44790578380800, 747955947110400, 13937735643955200, 287117441217331200, 5838778006909747200, 120976472421826560000, 2712639152754878054400
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OFFSET
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0,4
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COMMENTS
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a(n) is the number of n-permutations composed only of odd cycles of distinct length. - Geoffrey Critzer, Mar 08 2013
Also the number of permutations p of [n] with unique (functional) square root, i.e., there exists a unique permutation g such that g^2 = p. - Keith J. Bauer, Jan 08 2024
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LINKS
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FORMULA
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E.g.f.: Product_{m >= 1} (1+x^(2*m-1)/(2*m-1)). - Vladeta Jovovic, Nov 05 2003
a(n) ~ exp(-gamma/2) * n! / sqrt(2*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019
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MAPLE
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b:= proc(n, i) option remember; `if`(((i+1)/2)^2<n, 0,
`if`(n=0, 1, b(n, i-2)+`if`(i>n, 0, (i-1)!*
b(n-i, i-2)*binomial(n, i))))
end:
a:= n-> b(n, n-1+irem(n, 2)):
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MATHEMATICA
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nn=20; Range[0, nn]!CoefficientList[Series[Product[1+x^(2i-1)/(2i-1), {i, 1, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Mar 08 2013 *)
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PROG
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(PARI) {a(n)=n!*polcoeff( prod(k=1, n, 1+(k%2)*x^k/k, 1+x*O(x^n)), n)} /* Michael Somos, Sep 19 2006 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 01 2003
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EXTENSIONS
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STATUS
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approved
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