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A384368 Number of permutations of [2n] with n inversions.

%I A384368 #32 Jun 09 2025 06:18:41
%S A384368 1,1,5,29,174,1068,6655,41926,266338,1703027,10947079,70673825,
%T A384368 457927079,2976282415,19395654894,126688273871,829176461458,
%U A384368 5436687172806,35703722618623,234807844921153,1546217013188447,10193761267335877,67275841673522196,444431529264364506
%N A384368 Number of permutations of [2n] with n inversions.
%H A384368 Alois P. Heinz, <a href="/A384368/b384368.txt">Table of n, a(n) for n = 0..1208</a>
%F A384368 a(n) = A008302(2n,n).
%F A384368 a(n) ~ c * 3^(3*n - 1/2) / (sqrt(Pi*n) * 2^(2*n)), where c = QPochhammer(1/3) = A100220 = 0.5601260779279489449697922433141400143797363337983... - _Vaclav Kotesovec_, Jun 09 2025
%e A384368 a(0) = 1: the empty permutation.
%e A384368 a(1) = 1: 21.
%e A384368 a(2) = 5: 1342, 1423, 2143, 2314, 3124.
%e A384368 a(3) = 29: 123654, 124563, 124635, 125364, 125436, 126345, 132564, 132645, 134265, 134526, 135246, 142365, 142536, 143256, 152346, 213564, 213645, 214365, 214536, 215346, 231465, 231546, 234156, 241356, 312465, 312546, 314256, 321456, 412356.
%p A384368 a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..2*n), q, n+1), q, n):
%p A384368 seq(a(n), n=0..23);
%Y A384368 Cf. A008302, A128566.
%Y A384368 Cf. A100220.
%K A384368 nonn
%O A384368 0,3
%A A384368 _Alois P. Heinz_, May 27 2025