A285991 -id:A285991 - cp's OEIS frontend

A285675 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^n in powers of x.

1, -2, -2, 0, 6, 8, 4, -4, -18, -34, -32, -8, 36, 96, 144, 152, 94, -60, -294, -560, -760, -760, -460, 228, 1276, 2486, 3576, 4080, 3456, 1304, -2576, -7956, -13986, -19208, -21644, -19056, -9462, 8200, 33364, 63224, 92384, 112860, 114976, 88896, 26660, -74792

Offset: 0

Seiichi Manyama, Apr 30 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{n>0} ((1-x^n)/(1+x^n))^(n^m): A002448 (m=0), this sequence (m=1), A285988 (m=2), A285990 (m=3), A285991 (m=4).

Cf. A076577, A156616.

a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A076577(k)*a(n-k) for n > 0.
G.f.: exp(Sum_{k>=1} (sigma_2(k) - sigma_2(2*k))*x^k/(2*k)). - Ilya Gutkovskiy, Apr 14 2019
G.f.: exp( - 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^2) ). Cf. A000122. - Peter Bala, Dec 23 2021

A285990 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^(n^3) in powers of x.

Original entry on oeis.org

1, -2, -14, -24, 78, 536, 1236, -308, -12322, -45218, -73680, 76144, 872868, 2833904, 4612952, -2467592, -42205746, -147191388, -285572658, -127256088, 1376616024, 6138841704, 14949184532, 19201535108, -18287313476, -186761626394, -604980766280

Offset: 0

Seiichi Manyama, Apr 30 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..5987

Product_{n>0} ((1-x^n)/(1+x^n))^(n^m): A002448 (m=0), A285675 (m=1), A285988 (m=2), this sequence (m=3), A285991 (m=4).

Cf. A206623, A285989.

a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A285989(k)*a(n-k) for n > 0.

G.f.: exp(Sum_{k>=1} (sigma_4(k) - sigma_4(2*k))*x^k/(8*k)). - Ilya Gutkovskiy, Apr 14 2019

A285988 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^(n^2) in powers of x.

Original entry on oeis.org

1, -2, -6, -4, 22, 72, 92, -48, -522, -1294, -1624, 300, 6948, 19032, 30192, 20432, -45578, -202788, -437178, -599460, -311112, 1038624, 4023532, 8423280, 11892004, 8429270, -12073032, -60747944, -139842736, -223644552, -232762256, -15050944, 636838518

Offset: 0

Seiichi Manyama, Apr 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{n>0} ((1-x^n)/(1+x^n))^(n^m): A002448 (m=0), A285675 (m=1), this sequence (m=2), A285990 (m=3), A285991 (m=4).

Cf. A007331, A206622.

a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A007331(k)*a(n-k) for n > 0.

G.f.: exp(Sum_{k>=1} (sigma_3(k) - sigma_3(2*k))*x^k/(4*k)). - Ilya Gutkovskiy, Apr 14 2019

cp's OEIS Frontend

A285675 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^n in powers of x.

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A285990 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^(n^3) in powers of x.

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A285988 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^(n^2) in powers of x.

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