A320972 -id:A320972 - cp's OEIS frontend

A320971 Expansion of Product_{k>=1} ((1 - x^k)/(1 + x^k))^(sigma(k)).

1, -2, -4, 2, 10, 22, -4, -26, -68, -104, -12, 110, 378, 486, 448, -66, -1130, -2242, -3044, -2474, -322, 5106, 11064, 16954, 17896, 10440, -8032, -40132, -74578, -105754, -108564, -66534, 42672, 209858, 421352, 611946, 690204, 553534, 82112, -735082, -1892200

Offset: 0

Seiichi Manyama, Oct 25 2018

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

Convolution inverse of A301555.
Product_{k>=1} ((1 - x^k)/(1 + x^k))^(sigma_b(k)): A320908 (b=0), this sequence (b=1), A320972 (b=2).
Cf. A000203, A288385, A288421.

m:=80; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(  (&*[((1-q^k)/(1+q^k))^DivisorSigma(1,k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

With[{nmax=80}, CoefficientList[Series[Product[((1-q^k)/(1+q^k) )^DivisorSigma[1,k], {k, 1, nmax+2}], {q, 0, nmax}], q]] (* G. C. Greubel, Oct 29 2018 *)

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^k)/(1+x^k))^sigma(k)))

A321068 a(n) = [x^n] Product_{k>=1} ((1 - x^k)/(1 + x^k))^sigma_n(k).

Original entry on oeis.org

1, -2, -8, -22, 294, 24982, 1372372, 10145326, -38651841784, -21995644478504, -5088041946350856, 29713279339187796814, 155715351422115081062330, 370606511915720675179342334, -12360092915168107023209454901320

Offset: 0

Seiichi Manyama, Oct 26 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A320908, A320971, A320972, A321042, A321057.

Table[SeriesCoefficient[Product[((1 - x^k)/(1 + x^k))^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 27 2018 *)

{a(n) = polcoeff(prod(k=1, n, ((1-x^k+x*O(x^n))/(1+x^k+x*O(x^n)))^sigma(k, n)), n)}

cp's OEIS Frontend

A320971 Expansion of Product_{k>=1} ((1 - x^k)/(1 + x^k))^(sigma(k)).

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