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

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

0,5

Comments

Row sums equal A010790(n) = n!*(n+1)! for n>=0. Central terms form a bisection of A127728. Dual triangle is A129276.

Links

Table of n, a(n) for n=0..40.

Eric Weisstein's World of Mathematics, q-Factorial.

Formula

T(n,k) = [q^(nk+k)] Product_{i=1..n+1} { (1-q^i)/(1-q) }^2.

Example

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; ...

Prog

(PARI) T(n, k)=polcoeff(prod(i=1, n+1, (1-x^i)/(1-x))^2, (n+1)*k)

Crossrefs

Cf. A129275 (column 1); A127728 (central terms), A010790 (row sums); related triangles: A129276, A128564, A008302 (Mahonian numbers).

Sequence in context: A152971 A142459 A157641 * A176021 A166972 A364071

Adjacent sequences: A129271 A129272 A129273 * A129275 A129276 A129277

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 07 2007

Status

approved