A129278
Column 2 of triangle A129276; a(n) is the coefficient of q^(2n+2) in the squared q-factorial of n+2.
Original entry on oeis.org
1, 8, 106, 1558, 23589, 360499, 5530445, 85040656, 1310211074, 20223812711, 312728963211, 4844238210807, 75161494829564, 1167972975649534, 18175688264003359, 283219807758089612, 4418626917709277371
Offset: 0
A129277
Column 1 of triangle A129276; a(n) is the coefficient of q^n in the squared q-factorial of n+1.
Original entry on oeis.org
1, 2, 8, 42, 241, 1444, 8867, 55320, 349009, 2220242, 14215521, 91487834, 591285123, 3834960060, 24947236547, 162704291214, 1063516446543, 6965286759424, 45696734431169, 300262228345720, 1975679169075314
Offset: 0
A129274
Triangle, read by rows, where T(n,k) is the coefficient of q^(nk+k) in the squared q-factorial of n+1.
Original entry on oeis.org
1, 1, 1, 1, 10, 1, 1, 71, 71, 1, 1, 474, 1930, 474, 1, 1, 3103, 40096, 40096, 3103, 1, 1, 20190, 739929, 2108560, 739929, 20190, 1, 1, 131204, 12836959, 88638236, 88638236, 12836959, 131204, 1, 1, 853176, 215022825, 3286786158, 7625997280
Offset: 0
Definition of q-factorial of n:
faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
Obtain row 3 from coefficients in the squared q-factorial of 4:
faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
= (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
the resulting coefficients of q are:
[(1), 6, 19, 42, (71), 96, 106, 96, (71), 42, 19, 6, (1)],
where the terms enclosed in parenthesis form row 3.
Triangle begins:
1;
1, 1;
1, 10, 1;
1, 71, 71, 1;
1, 474, 1930, 474, 1;
1, 3103, 40096, 40096, 3103, 1;
1, 20190, 739929, 2108560, 739929, 20190, 1;
1, 131204, 12836959, 88638236, 88638236, 12836959, 131204, 1;
1, 853176, 215022825, 3286786158, 7625997280, 3286786158, 215022825, 853176, 1; ...
-
T(n,k)=polcoeff(prod(i=1,n+1,(1-x^i)/(1-x))^2,(n+1)*k)
Showing 1-3 of 3 results.
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