A128597
Column 2 of triangle A128596; a(n) = coefficient of q^(2n+4) in the q-analog of the even double factorials (2n+4)!! for n>=0.
Original entry on oeis.org
1, 7, 46, 297, 1919, 12399, 80241, 520399, 3382588, 22034519, 143826980, 940569228, 6161492611, 40426009162, 265617089899, 1747501590554, 11510584144337, 75901841055650, 501007227527884, 3310076954166501
Offset: 0
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{a(n)=polcoeff(prod(j=1,n+2,(1-q^(2*j))/(1-q)),2*n+4,q)}
A128598
Column 3 of triangle A128596; a(n) = coefficient of q^(3n+9) in the q-analog of the even double factorials (2n+6)!! for n>=0.
Original entry on oeis.org
1, 24, 297, 3210, 32510, 318171, 3054100, 28980565, 273077443, 2562036673, 23973009386, 223949654108, 2090070431683, 19496003736658, 181815760387221, 1695523268254637, 15813185728272754, 147508341317700463
Offset: 0
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{a(n)=polcoeff(prod(j=1,n+3,(1-q^(2*j))/(1-q)),3*n+9,q)}
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.
Original entry on oeis.org
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
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;
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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 *)
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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)))
Showing 1-3 of 3 results.