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)}
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)}
Original entry on oeis.org
1, 2, 5, 26, 189, 1734, 19305, 253370, 3828825, 65473006, 1249937325, 26352843470, 608142583125, 15247003381854, 412685556908625, 11993673995924378, 372509404162520625, 12313505304343363126, 431620764875678503125
Offset: 0
-
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 *)
-
{a(n)=sum(k=0,n,polcoeff(prod(j=1,n+1,(1-q^(2*j-1))/(1-q)),(n+1)*k,q))}
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.
Original entry on oeis.org
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
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;
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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)))
Showing 1-4 of 4 results.
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