A128593
Column 1 of triangle A128592; a(n) = coefficient of q^(n+2) in the q-analog of the odd double factorials (2n+3)!! for n>=0.
Original entry on oeis.org
1, 3, 12, 45, 170, 644, 2451, 9365, 35908, 138104, 532589, 2058782, 7975216, 30951921, 120326060, 468473348, 1826415556, 7129330988, 27860219331, 108984557708, 426730087879, 1672310507262, 6558840830680, 25742937514814, 101108341344396, 397368218111003
Offset: 0
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b:= proc(n) option remember; `if`(n=0, 1,
simplify(b(n-1)*(1-q^(2*n-1))/(1-q)))
end:
a:= n-> coeff(b(n+2), q, n+2):
seq(a(n), n=0..30); # Alois P. Heinz, Sep 22 2021
-
a[n_] := SeriesCoefficient[Product[(1-q^(2j-1))/(1-q), {j, 1, n+2}], {q, 0, n+2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 27 2024 *)
-
{a(n)=polcoeff(prod(j=1,n+2,(1-q^(2*j-1))/(1-q)),n+2,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;
-
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 *)
-
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)))
A128594
Column 2 of triangle A128592; a(n) = coefficient of q^(2n+6) in the q-analog of the odd double factorials (2n+5)!! for n>=0.
Original entry on oeis.org
1, 12, 97, 696, 4784, 32230, 214978, 1426566, 9441417, 62405645, 412278981, 2723566163, 17996243101, 118957645301, 786700165122, 5205396517853, 34461624895701, 228274455988134, 1512920531980961, 10032446308837778
Offset: 0
-
a[n_] := SeriesCoefficient[Product[(1-q^(2j-1))/(1-q), {j, 1, n+3}], {q, 0, 2n+6}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 27 2024 *)
-
{a(n)=polcoeff(prod(j=1,n+3,(1-q^(2*j-1))/(1-q)),2*n+6,q)}
Showing 1-3 of 3 results.