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A363180 Number of permutations of [2n] with n parity changes.

%I A363180 #38 May 26 2023 08:18:50
%S A363180 1,2,8,288,10368,1036800,103680000,20321280000,3982970880000,
%T A363180 1290482565120000,418116351098880000,202368313931857920000,
%U A363180 97946263943019233280000,66211674425481001697280000,44759091911625157147361280000,40283182720462641432625152000000
%N A363180 Number of permutations of [2n] with n parity changes.
%H A363180 Alois P. Heinz, <a href="/A363180/b363180.txt">Table of n, a(n) for n = 0..225</a>
%F A363180 a(n) = A152874(2n,n).
%F A363180 From _Vaclav Kotesovec_, May 26 2023: (Start)
%F A363180 Recurrence: (2*n - 3)*a(n) = 4*(2*n^2 - 4*n + 1)*a(n-1) + 16*(n-2)^2*(n-1)^2*(2*n - 1)*a(n-2).
%F A363180 a(n) ~ 2^(2*n+1) * n^(2*n) / exp(2*n). (End)
%e A363180 a(0) = 1: (), the empty permutation.
%e A363180 a(1) = 2: 12, 21.
%e A363180 a(2) = 8: 1243, 1423, 2134, 2314, 3241, 3421, 4132, 4312.
%e A363180 a(3) = 288: 123546, 123564, 124356, 124536, 125346, ..., 652431, 653241, 653421, 654213, 654231.
%p A363180 a:= proc(n) option remember; `if`(n<2, 2^n,
%p A363180       (16*(n-2)^2*(2*n-1)*(n-1)^2*a(n-2)+4*(2*n^2-4*n+1)*a(n-1))/(2*n-3))
%p A363180     end:
%p A363180 seq(a(n), n=0..18);
%Y A363180 Cf. A152874, A363236.
%K A363180 nonn
%O A363180 0,2
%A A363180 _Alois P. Heinz_, May 23 2023