A128596 Triangle, read by rows, of coefficients of q^(nk) in the q-analog of the even double factorials: T(n,k) = [q^(nk)] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.

1, 1, 1, 1, 2, 1, 1, 7, 7, 1, 1, 24, 46, 24, 1, 1, 86, 297, 297, 86, 1, 1, 315, 1919, 3210, 1919, 315, 1, 1, 1170, 12399, 32510, 32510, 12399, 1170, 1, 1, 4389, 80241, 318171, 484636, 318171, 80241, 4389, 1, 1, 16588, 520399, 3054100, 6730832, 6730832

Offset

0,5

Links

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

Formula

T(n,k) = A128084(n,nk) where A128084 is the triangle of coefficients of q in the q-analog of the even double factorials.

Example

Row sums equal 2*A000165(n-1) for n>0, twice the even double factorials:
[1, 2, 4, 16, 96, 768, 7680, 92160, 1290240, ..., 2*(2n-2)!!, ...].
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 7, 7, 1;
1, 24, 46, 24, 1;
1, 86, 297, 297, 86, 1;
1, 315, 1919, 3210, 1919, 315, 1;
1, 1170, 12399, 32510, 32510, 12399, 1170, 1;
1, 4389, 80241, 318171, 484636, 318171, 80241, 4389, 1;
1, 16588, 520399, 3054100, 6730832, 6730832, 3054100, 520399, 16588, 1;
1, 63064, 3382588, 28980565, 89514691, 127707302, 89514691, 28980565, 3382588, 63064, 1;

Prog

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

Crossrefs

Cf. A128084; A000165 ((2n)!!); A128086 (column 1), A128597 (column 2), A128598 (column 3); variant: A128592.

Sequence in context: A220602 A166345 A015110 * A176305 A139349 A168347

Adjacent sequences: A128593 A128594 A128595 * A128597 A128598 A128599

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 12 2007

Status

approved