cp's OEIS Frontend

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

%I A127728 #27 Feb 16 2025 08:33:04
%S A127728 1,1,2,10,106,1930,53612,2108560,111482424,7625997280,655331699940,
%T A127728 69110082376388,8775534280695310,1320693932817784342,
%U A127728 232459627389638257316,47311901973588298051380,11025565860152700884475938,2916827988004938784779055448
%N A127728 Sum of squared coefficients of q in the q-factorials.
%C A127728 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
%H A127728 Vaclav Kotesovec, <a href="/A127728/b127728.txt">Table of n, a(n) for n = 0..100</a>
%H A127728 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%F A127728 Conjecture: a(n) ~ 6 * sqrt(Pi) * n^(2*n - 1/2) / exp(2*n). - _Vaclav Kotesovec_, Oct 22 2020
%F A127728 a(n) = Sum_{k>=0} A008302(n,k)^2. - _R. J. Mathar_, Jan 06 2022
%e A127728 Definition of q-factorial of n:
%e A127728 faq(n) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0)=1;
%e A127728 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;
%e A127728 then a(n) is the sum of squared coefficients of q:
%e A127728 a(4) = 1^2 + 3^2 + 5^2 + 6^2 + 5^2 + 3^2 + 1^2 = 106.
%t A127728 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 *)
%o A127728 (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)}
%o A127728 (PARI) a(n) = norml2(Vec(prod(k=1, n, (1-q^k)/(1-q)))); \\ _Michel Marcus_, Jan 18 2025
%Y A127728 Cf. A008302, A380274, A380275.
%K A127728 nonn
%O A127728 0,3
%A A127728 _Paul D. Hanna_, Jan 25 2007