A129276 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, 2, 1, 1, 8, 8, 1, 1, 42, 106, 42, 1, 1, 241, 1558, 1558, 241, 1, 1, 1444, 23589, 53612, 23589, 1444, 1, 1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1, 1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1

Offset

0,5

Comments

Dual triangle is A129274.

Central terms form a bisection of A127728.

Links

Table of n, a(n) for n=0..44.

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

Formula

T(n,k) = [q^(nk-k)] Product_{i=1..n} { (1-q^i)/(1-q) }^2 for n>0, with T(0,0)=1.

Row sums = (n!)^2/(n-1) for n>=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 4 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 4.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 8, 8, 1;
1, 42, 106, 42, 1;
1, 241, 1558, 1558, 241, 1;
1, 1444, 23589, 53612, 23589, 1444, 1;
1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1;
1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1; ...

Mathematica

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 *)

Prog

(PARI) T(n, k)=if(n==0, 1, polcoeff(prod(i=1, n, (1-x^i)/(1-x))^2, (n-1)*k))

Crossrefs

Cf. A129277 (column 1), A129278 (column 2); A127728 (central terms), related triangles: A129274, A128564, A008302 (Mahonian numbers).

Sequence in context: A229922 A051428 A176698 * A156901 A167400 A225848

Adjacent sequences: A129273 A129274 A129275 * A129277 A129278 A129279

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 07 2007

Status

approved