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A129274 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk+k) in the squared q-factorial of n+1.

%I A129274 #8 Feb 16 2025 08:33:05
%S A129274 1,1,1,1,10,1,1,71,71,1,1,474,1930,474,1,1,3103,40096,40096,3103,1,1,
%T A129274 20190,739929,2108560,739929,20190,1,1,131204,12836959,88638236,
%U A129274 88638236,12836959,131204,1,1,853176,215022825,3286786158,7625997280
%N A129274 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk+k) in the squared q-factorial of n+1.
%C A129274 Row sums equal A010790(n) = n!*(n+1)! for n>=0. Central terms form a bisection of A127728. Dual triangle is A129276.
%H A129274 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%F A129274 T(n,k) = [q^(nk+k)] Product_{i=1..n+1} { (1-q^i)/(1-q) }^2.
%e A129274 Definition of q-factorial of n:
%e A129274 faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
%e A129274 Obtain row 3 from coefficients in the squared q-factorial of 4:
%e A129274 faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
%e A129274 = (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
%e A129274 the resulting coefficients of q are:
%e A129274 [(1), 6, 19, 42, (71), 96, 106, 96, (71), 42, 19, 6, (1)],
%e A129274 where the terms enclosed in parenthesis form row 3.
%e A129274 Triangle begins:
%e A129274 1;
%e A129274 1, 1;
%e A129274 1, 10, 1;
%e A129274 1, 71, 71, 1;
%e A129274 1, 474, 1930, 474, 1;
%e A129274 1, 3103, 40096, 40096, 3103, 1;
%e A129274 1, 20190, 739929, 2108560, 739929, 20190, 1;
%e A129274 1, 131204, 12836959, 88638236, 88638236, 12836959, 131204, 1;
%e A129274 1, 853176, 215022825, 3286786158, 7625997280, 3286786158, 215022825, 853176, 1; ...
%o A129274 (PARI) T(n,k)=polcoeff(prod(i=1,n+1,(1-x^i)/(1-x))^2,(n+1)*k)
%Y A129274 Cf. A129275 (column 1); A127728 (central terms), A010790 (row sums); related triangles: A129276, A128564, A008302 (Mahonian numbers).
%K A129274 nonn,tabl
%O A129274 0,5
%A A129274 _Paul D. Hanna_, Apr 07 2007