A128595 Row sums of triangle A128592.

1, 2, 5, 26, 189, 1734, 19305, 253370, 3828825, 65473006, 1249937325, 26352843470, 608142583125, 15247003381854, 412685556908625, 11993673995924378, 372509404162520625, 12313505304343363126, 431620764875678503125

Offset: 0

Author

Paul D. Hanna, Mar 12 2007

Keywords

• nonn

Comments

A128592(n,k) is the coefficient of q^(nk+k) in the q-analog of the odd double factorials (2n-1)!!.

Crossrefs

Cf. A128592; A128080; A001147 ((2n-1)!!); A128593 (column 1), A128594 (column 2).

Programs

Mathematica

a[n_] := Sum[SeriesCoefficient[Product[(1-q^(2j-1))/(1-q), {j, 1, n+1}], {q, 0, k(n+1)}], {k, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 27 2024 *)

PARI

{a(n)=sum(k=0,n,polcoeff(prod(j=1,n+1,(1-q^(2*j-1))/(1-q)),(n+1)*k,q))}

Formula

a(n) = Sum_{k=0..n} { [q^(nk+k)] Product_{j=1..n+1} (1-q^(2j-1))/(1-q) } for n>=0.