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A088994 Number of permutations in the symmetric group S_n such that the size of their centralizer is odd.

%I A088994 #47 Jan 09 2024 08:50:30
%S A088994 1,1,0,2,8,24,144,720,8448,64512,576000,5529600,74972160,887546880,
%T A088994 11285084160,168318259200,2843121254400,44790578380800,
%U A088994 747955947110400,13937735643955200,287117441217331200,5838778006909747200,120976472421826560000,2712639152754878054400
%N A088994 Number of permutations in the symmetric group S_n such that the size of their centralizer is odd.
%C A088994 a(n) is the number of n-permutations composed only of odd cycles of distinct length. - _Geoffrey Critzer_, Mar 08 2013
%C A088994 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
%H A088994 Seiichi Manyama, <a href="/A088994/b088994.txt">Table of n, a(n) for n = 0..450</a>
%F A088994 E.g.f.: Product_{m >= 1} (1+x^(2*m-1)/(2*m-1)). - _Vladeta Jovovic_, Nov 05 2003
%F A088994 a(n) ~ exp(-gamma/2) * n! / sqrt(2*n), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jul 23 2019
%F A088994 a(n) = n! - A088335(n). - _Alois P. Heinz_, Jan 27 2020
%p A088994 b:= proc(n, i) option remember; `if`(((i+1)/2)^2<n, 0,
%p A088994       `if`(n=0, 1, b(n,i-2)+`if`(i>n, 0, (i-1)!*
%p A088994        b(n-i, i-2)*binomial(n, i))))
%p A088994     end:
%p A088994 a:= n-> b(n, n-1+irem(n, 2)):
%p A088994 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 01 2017
%t A088994 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 *)
%o A088994 (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 */
%Y A088994 Cf. A000142, A007838, A007841, A087639, A088335, A294506, A305199, A309319.
%K A088994 nonn
%O A088994 0,4
%A A088994 Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 01 2003
%E A088994 More terms from _Vladeta Jovovic_, Nov 03 2003
%E A088994 a(0)=1 prepended by _Seiichi Manyama_, Nov 01 2017