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%I #9 Apr 08 2020 07:53:22
%S 0,0,2,4,16,96,576,4320,31872,298368,3052800,34387200,404029440,
%T 5339473920,75893207040,1139356108800,18079668633600,310896849715200,
%U 5654417758617600,107707364764876800,2145784566959308800,45252164164799692800,1003024255355781120000
%N Number of permutations in the symmetric group S_n such that the size of their centralizer is even.
%H Andrew Howroyd, <a href="/A088335/b088335.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = n! - A088994(n).
%p b:= proc(n, i) option remember; `if`(((i+1)/2)^2<n, 0,
%p `if`(n=0, 1, b(n, i-2)+`if`(i>n, 0, (i-1)!*
%p b(n-i, i-2)*binomial(n, i))))
%p end:
%p a:= n-> n!-b(n, n-1+irem(n, 2)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 27 2020
%t b[n_, i_] := b[n, i] = 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]]]];
%t a[n_] := n! - b[n, n - 1 + Mod[n, 2]];
%t a /@ Range[0, 30] (* _Jean-François Alcover_, Apr 08 2020, after _Alois P. Heinz_ *)
%o (PARI) seq(n)={Vec(serlaplace(1/(1-x) - prod(k=1, n, 1+(k%2)*x^k/k + O(x*x^n))), -(n+1))} \\ _Andrew Howroyd_, Jan 27 2020
%Y Cf. A000142, A088994.
%K nonn
%O 0,3
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 07 2003
%E a(0)=0 prepended and terms a(11) and beyond from _Andrew Howroyd_, Jan 27 2020
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