A127728 Sum of squared coefficients of q in the q-factorials.

1, 1, 2, 10, 106, 1930, 53612, 2108560, 111482424, 7625997280, 655331699940, 69110082376388, 8775534280695310, 1320693932817784342, 232459627389638257316, 47311901973588298051380, 11025565860152700884475938, 2916827988004938784779055448

Offset: 0

Author

Paul D. Hanna, Jan 25 2007

Keywords

• nonn

Comments

Two n-permutations are randomly selected from S_n with replacement. a(n)/(n!)^2 is the probability that they will have the same number of inversions. - Geoffrey Critzer, May 15 2010

Examples

			Definition of q-factorial of n:
faq(n) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0)=1;
faq(4) = 1*(1 + q)*(1 + q + q^2)*(1 + q + q^2 + q^3) = 1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6;
then a(n) is the sum of squared coefficients of q:
a(4) = 1^2 + 3^2 + 5^2 + 6^2 + 5^2 + 3^2 + 1^2 = 106.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..100
• Eric Weisstein's World of Mathematics, q-Factorial.

Crossrefs

Cf. A008302, A380274, A380275.

Programs

Mathematica

Table[Total[ CoefficientList[Expand[Product[Sum[x^i, {i, 0, m}], {m, 1, n - 1}]], x]^2], {n, 0, 15}] (* Geoffrey Critzer, May 15 2010 *)

PARI

{a(n)=local(faq_n=if(n==0,1,prod(k=1,n,(1-q^k)/(1-q)))); sum(k=0,n*(n-1)/2,polcoeff(faq_n,k,q)^2)}

PARI

a(n) = norml2(Vec(prod(k=1, n, (1-q^k)/(1-q)))); \\ Michel Marcus, Jan 18 2025

Formula

Conjecture: a(n) ~ 6 * sqrt(Pi) * n^(2*n - 1/2) / exp(2*n). - Vaclav Kotesovec, Oct 22 2020

a(n) = Sum_{k>=0} A008302(n,k)^2. - R. J. Mathar, Jan 06 2022