A128592 Triangle, read by rows, of coefficients of q^(nk+k) in the q-analog of the odd double factorials: T(n,k) = [q^(nk+k)] Product_{j=1..n+1} (1-q^(2j-1))/(1-q) for n>0, with T(0,0)=1.

1, 1, 1, 1, 3, 1, 1, 12, 12, 1, 1, 45, 97, 45, 1, 1, 170, 696, 696, 170, 1, 1, 644, 4784, 8447, 4784, 644, 1, 1, 2451, 32230, 92003, 92003, 32230, 2451, 1, 1, 9365, 214978, 946330, 1487477, 946330, 214978, 9365, 1, 1, 35908, 1426566, 9417798, 21856230, 21856230

Offset

0,5

Links

Table of n, a(n) for n=0..50.

Formula

T(n,k) = A128080(n+1,nk+k) where A128080 is the triangle of coefficients of q in the q-analog of the odd double factorials.

Example

Triangle begins:
1;
1, 1;
1, 3, 1;
1, 12, 12, 1;
1, 45, 97, 45, 1;
1, 170, 696, 696, 170, 1;
1, 644, 4784, 8447, 4784, 644, 1;
1, 2451, 32230, 92003, 92003, 32230, 2451, 1;
1, 9365, 214978, 946330, 1487477, 946330, 214978, 9365, 1;
1, 35908, 1426566, 9417798, 21856230, 21856230, 9417798, 1426566, 35908, 1;
1, 138104, 9441417, 91852376, 302951392, 441170745, 302951392, 91852376, 9441417, 138104, 1;

Mathematica

T[n_, k_] := If[k < 0 || k > n*(n + 1), 0, If[n == 0, 1, SeriesCoefficient[Product[(1 - q^(2*j - 1))/(1 - q), {j, 1, n + 1}], {q, 0, (n + 1)*k}]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 27 2022, from PARI code *)

Prog

(PARI) T(n, k)=if(k<0 || k>n*(n+1), 0, if(n==0, 1, polcoeff(prod(j=1, n+1, (1-q^(2*j-1))/(1-q)), (n+1)*k, q)))

Crossrefs

Cf. A128080; A001147 ((2n-1)!!); A128593 (column 1), A128594 (column 2), A128595 (row sums); variant: A128596.

Sequence in context: A134523 A098778 A078122 * A156584 A209424 A129619

Adjacent sequences: A128589 A128590 A128591 * A128593 A128594 A128595

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 12 2007

Status

approved