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A129276 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk-k) in the squared q-factorial of n.

%I A129276 #16 Feb 16 2025 08:33:05
%S A129276 1,1,1,1,2,1,1,8,8,1,1,42,106,42,1,1,241,1558,1558,241,1,1,1444,23589,
%T A129276 53612,23589,1444,1,1,8867,360499,1747433,1747433,360499,8867,1,1,
%U A129276 55320,5530445,54794622,111482424,54794622,5530445,55320,1
%N A129276 Triangle, read by rows, where T(n,k) is the coefficient of q^(nk-k) in the squared q-factorial of n.
%C A129276 Dual triangle is A129274.
%C A129276 Central terms form a bisection of A127728.
%H A129276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%F A129276 T(n,k) = [q^(nk-k)] Product_{i=1..n} { (1-q^i)/(1-q) }^2 for n>0, with T(0,0)=1.
%F A129276 Row sums = (n!)^2/(n-1) for n>=2.
%e A129276 Definition of q-factorial of n:
%e A129276 faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
%e A129276 Obtain row 4 from coefficients in the squared q-factorial of 4:
%e A129276 faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
%e A129276 = (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
%e A129276 the resulting coefficients of q are:
%e A129276 [(1), 6, 19, (42), 71, 96, (106), 96, 71, (42), 19, 6, (1)],
%e A129276 where the terms enclosed in parenthesis form row 4.
%e A129276 Triangle begins:
%e A129276 1;
%e A129276 1, 1;
%e A129276 1, 2, 1;
%e A129276 1, 8, 8, 1;
%e A129276 1, 42, 106, 42, 1;
%e A129276 1, 241, 1558, 1558, 241, 1;
%e A129276 1, 1444, 23589, 53612, 23589, 1444, 1;
%e A129276 1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1;
%e A129276 1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1; ...
%t A129276 faq[n_, q_] := Product[(1-q^k)/(1-q), {k, 1, n}]; t[0, 0] = t[1, 0] = t[1, 1] = 1; t[n_, k_] := SeriesCoefficient[faq[n, q]^2, {q, 0, (n-1)*k}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 26 2013 *)
%o A129276 (PARI) T(n,k)=if(n==0,1,polcoeff(prod(i=1,n,(1-x^i)/(1-x))^2,(n-1)*k))
%Y A129276 Cf. A129277 (column 1), A129278 (column 2); A127728 (central terms), related triangles: A129274, A128564, A008302 (Mahonian numbers).
%K A129276 nonn,tabl
%O A129276 0,5
%A A129276 _Paul D. Hanna_, Apr 07 2007