A285675 -id:A285675 - cp's OEIS frontend

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

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

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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), 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

A285991 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^(n^4) in powers of x.

Original entry on oeis.org

1, -2, -30, -100, 262, 3672, 13836, -80, -264810, -1421438, -3019032, 7630764, 89648580, 358974280, 548677872, -2390377936, -20531491146, -74635378020, -110275527170, 425036176572, 3669041188152, 13597190512480, 23995331740700, -45340748171760

Offset: 0

Seiichi Manyama, Apr 30 2017

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Product_{n>0} ((1-x^n)/(1+x^n))^(n^m): A002448 (m=0), A285675 (m=1), A285988 (m=2), A285990 (m=3), this sequence (m=4).

Cf. A096960, A206624.

a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A096960(k)*a(n-k) for n > 0.

G.f.: exp(Sum_{k>=1} (sigma_5(k) - sigma_5(2*k))*x^k/(16*k)). - Ilya Gutkovskiy, Apr 14 2019

A296048 Expansion of e.g.f. Product_{k>=1} ((1 - x^k)/(1 + x^k))^(1/k).

Original entry on oeis.org

1, -2, 2, -4, 32, -128, 496, -2336, 29312, -395776, 3194624, -21951488, 277270528, -4027191296, 38850203648, -739834458112, 19460560584704, -299971773661184, 3169121209090048, -51853341314514944, 1234704403684130816, -30653318499154788352, 658369600764729884672, -10809496145754051313664

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

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Cf. A001227, A028342, A028343, A054844, A156616, A168243, A285675, A294356, A295792.

a:=series(mul(((1-x^k)/(1+x^k))^(1/k),k=1..100),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[-2 Sum[Total[Mod[Divisors[k], 2] x^k]/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(-2*Sum_{k>=1} A001227(k)*x^k/k).

E.g.f.: exp(-Sum_{k>=1} A054844(k)*x^k/k).

cp's OEIS Frontend

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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A285991 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^(n^4) in powers of x.

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A296048 Expansion of e.g.f. Product_{k>=1} ((1 - x^k)/(1 + x^k))^(1/k).

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