0,5

Central terms equal A063075 (number of partitions of 2n^2 whose Ferrers-plot fits within a 2n X 2n box and cover an n X n box).

The triangle of q-binomial coefficients:

C_q(n,k) = [Product_{i=n-k+1..n}(1-q^i)]/[Product_{j=1..k}(1-q^j)]

begins:

1;

1, 1;

1, 1+q, 1;

1, 1+q+q^2, 1+q+q^2, 1;

1, 1+q+q^2+q^3, 1+q+2*q^2+q^3+q^4, 1+q+q^2+q^3, 1; ...

recurrence: C_q(n+1,k) = C_q(n,k-1) + q^k * C_q(n,k).

Element T(n,k) of this triangle equals the sum of the squares

of the coefficients of q in q-binomial coefficient C_q(n,k).

This triangle begins:

1;

1, 1;

1, 2, 1;

1, 3, 3, 1;

1, 4, 8, 4, 1;

1, 5, 16, 16, 5, 1;

1, 6, 29, 48, 29, 6, 1;

1, 7, 47, 119, 119, 47, 7, 1;

1, 8, 72, 256, 390, 256, 72, 8, 1;

1, 9, 104, 500, 1070, 1070, 500, 104, 9, 1;

1, 10, 145, 900, 2592, 3656, 2592, 900, 145, 10, 1;

1, 11, 195, 1525, 5674, 10762, 10762, 5674, 1525, 195, 11, 1;

1, 12, 256, 2456, 11483, 28160, 37834, 28160, 11483, 2456, 256, 12, 1;

The central terms equal A063075:

1, 2, 8, 48, 390, 3656, 37834, 417540, 4836452, 58130756, ...

(PARI) {T(n, k)=local(C_q=if(n==0|k==0, 1, prod(j=n-k+1, n, 1-q^j)/prod(j=1, k, 1-q^j))); sum(i=0, (n-k)*k, polcoeff(C_q, i)^2)}

Cf. A063075 (central terms); A125807, A125808, A125809 (row sums).

Adjacent sequences: A125803 A125804 A125805 this_sequence A125807 A125808 A125809

Sequence in context: A073134 A026692 A114202 this_sequence A099597 A123610 A059922

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 11 2006