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A128566 Number of permutations of {1..n} with n inversions. 7

%I

%S 1,0,0,1,5,22,90,359,1415,5545,21670,84591,330121,1288587,5032235,

%T 19664205,76893687,300895513,1178290263,4617369760,18106447251,

%U 71048746505,278966179936,1095987764828,4308300939450,16944940572831,66680029591816,262519664110588

%N Number of permutations of {1..n} with n inversions.

%H Alois P. Heinz, <a href="/A128566/b128566.txt">Table of n, a(n) for n = 0..1665</a>

%F a(n) = A008302(n,n) = coefficient of q^n in the q-factorial of n.

%F 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 a(n) ~ c * 2^(2*n-1) / sqrt(Pi*n), where c = A048651 = QPochhammer[1/2] = 0.28878809508660242127889972192923... . - _Vaclav Kotesovec_, Sep 07 2014

%p a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..n), q, n+1), q, n):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 05 2013

%t Table[SeriesCoefficient[QPochhammer[x, x, n]/(1-x)^n, {x, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, May 13 2016 *)

%o (PARI) {a(n)=polcoeff(prod(j=1, n, (1-q^j)/(1-q)),n,q)}

%Y Diagonal of A008302 (Mahonian numbers).

%Y Column 2 of A128564.

%Y Cf. A128565 (column 1), A214086, A048651.

%K nonn

%O 0,5

%A _Paul D. Hanna_, Mar 12 2007

%E Edited by _Alois P. Heinz_, Mar 05 2013

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Last modified March 2 09:23 EST 2021. Contains 341746 sequences. (Running on oeis4.)