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A128566 Number of permutations of {1..n} with n inversions.

%I A128566 #28 Aug 30 2020 19:30:09
%S A128566 1,0,0,1,5,22,90,359,1415,5545,21670,84591,330121,1288587,5032235,
%T A128566 19664205,76893687,300895513,1178290263,4617369760,18106447251,
%U A128566 71048746505,278966179936,1095987764828,4308300939450,16944940572831,66680029591816,262519664110588
%N A128566 Number of permutations of {1..n} with n inversions.
%H A128566 Alois P. Heinz, <a href="/A128566/b128566.txt">Table of n, a(n) for n = 0..1665</a>
%F A128566 a(n) = A008302(n,n) = coefficient of q^n in the q-factorial of n.
%F A128566 a(n) = T(n,n) with T(n,k) = T(n-1,k) + Sum_{j=1..n-1} T(n-1,k-j) for n>=0, k>0; T(n,k) = 0 for n<0; T(n,0) = 1 for n>=0. - _Alois P. Heinz_, Mar 07 2013
%F A128566 a(n) ~ c * 2^(2*n-1) / sqrt(Pi*n), where c = A048651 = QPochhammer[1/2] = 0.28878809508660242127889972192923... . - _Vaclav Kotesovec_, Sep 07 2014
%p A128566 a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..n), q, n+1), q, n):
%p A128566 seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 05 2013
%t A128566 Table[SeriesCoefficient[QPochhammer[x, x, n]/(1-x)^n, {x, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, May 13 2016 *)
%o A128566 (PARI) {a(n)=polcoeff(prod(j=1, n, (1-q^j)/(1-q)),n,q)}
%Y A128566 Diagonal of A008302 (Mahonian numbers).
%Y A128566 Column 2 of A128564.
%Y A128566 Cf. A128565 (column 1), A214086, A048651.
%K A128566 nonn
%O A128566 0,5
%A A128566 _Paul D. Hanna_, Mar 12 2007
%E A128566 Edited by _Alois P. Heinz_, Mar 05 2013