A363180 - OEIS

%I #38 May 26 2023 08:18:50

%S 1,2,8,288,10368,1036800,103680000,20321280000,3982970880000,

%T 1290482565120000,418116351098880000,202368313931857920000,

%U 97946263943019233280000,66211674425481001697280000,44759091911625157147361280000,40283182720462641432625152000000

%N Number of permutations of [2n] with n parity changes.

%H Alois P. Heinz, <a href="/A363180/b363180.txt">Table of n, a(n) for n = 0..225</a>

%F a(n) = A152874(2n,n).

%F From _Vaclav Kotesovec_, May 26 2023: (Start)

%F 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 a(n) ~ 2^(2*n+1) * n^(2*n) / exp(2*n). (End)

%e a(0) = 1: (), the empty permutation.

%e a(1) = 2: 12, 21.

%e a(2) = 8: 1243, 1423, 2134, 2314, 3241, 3421, 4132, 4312.

%e a(3) = 288: 123546, 123564, 124356, 124536, 125346, ..., 652431, 653241, 653421, 654213, 654231.

%p a:= proc(n) option remember; `if`(n<2, 2^n,

%p       (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     end:

%p seq(a(n), n=0..18);

%Y Cf. A152874, A363236.

%K nonn

%O 0,2

%A _Alois P. Heinz_, May 23 2023