0,5

Row sums = (n!)^2/(n-1) for n>=2. Central terms form a bisection of A127728. Dual triangle is A129274.

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

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

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; ...

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

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

Sequence in context: A157785 A021476 A051428 this_sequence A156901 A167400 A165889

Adjacent sequences: A129273 A129274 A129275 this_sequence A129277 A129278 A129279

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 07 2007