A275585 -id:A275585 - cp's OEIS frontend

A288389 Expansion of Product_{k>=1} (1 - x^k)^(sigma_2(k)).

1, -1, -5, -5, -1, 35, 66, 100, 15, -330, -841, -1591, -1468, 426, 6306, 16399, 27745, 31544, 6364, -70389, -225322, -435265, -617937, -537135, 176008, 1970213, 5150080, 9277624, 12631298, 11048049, -1884235, -34460900, -92385183, -171971785, -247790333

Offset: 0

Seiichi Manyama, Jun 08 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A027847.

Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), A288385 (m=1), this sequence (m=2), A288392 (m=3).

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-q^k)^DivisorSigma(2,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[2](d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 08 2017

nmax = 50; CoefficientList[Series[Product[(1-x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 30 2018 *)

m=50; x='x+O('x^m); Vec(prod(k=1, m, (1-x^k)^sigma(k,2))) \\ G. C. Greubel, Oct 30 2018

Convolution inverse of A275585.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A027847(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 29 2018

A301556 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma_2(k)).

Original entry on oeis.org

1, 2, 12, 42, 154, 498, 1640, 4990, 15092, 43840, 125220, 348478, 954294, 2561714, 6776404, 17644494, 45338734, 114971434, 288148860, 713968998, 1750662814, 4249685398, 10219662844, 24356466418, 57558783492, 134922807056, 313842321696, 724651728916

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Convolution of A275585 and A288414.

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A001157, A275585, A288414.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(5/4) * Pi * Zeta(3)^(1/4) * n^(3/4)/3 - Pi*n^(1/4) / (3 * 2^(13/4) * Zeta(3)^(1/4)) + Zeta(3)/(8*Pi^2)) * Zeta(3)^(1/8) / (2^(15/8) * n^(5/8)).

A321876 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - x^j)^sigma_k(j).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 6, 8, 11, 1, 1, 10, 16, 21, 17, 1, 1, 18, 38, 52, 39, 34, 1, 1, 34, 100, 156, 128, 92, 52, 1, 1, 66, 278, 526, 534, 373, 170, 94, 1, 1, 130, 796, 1896, 2546, 2014, 913, 360, 145, 1, 1, 258, 2318, 7102, 13074, 12953, 6796, 2399, 667, 244

Offset: 0

Ilya Gutkovskiy, Nov 20 2018

			Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   1,   1,    1,    1,     1,      1,  ...
   3,   4,    6,   10,    18,     34,  ...
   5,   8,   16,   38,   100,    278,  ...
  11,  21,   52,  156,   526,   1896,  ...
  17,  39,  128,  534,  2546,  13074,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..9 give A006171, A061256, A275585, A288391, A301542, A301543, A301544, A301545, A301546, A301547.
Main diagonal gives A319647.
Cf. A321877.

Table[Function[k, SeriesCoefficient[Product[1/(1 - x^j)^DivisorSigma[k, j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[DivisorSigma[k + 1, j] x^j/(j (1 - x^j)), {j, 1, n}]], {x, 0, n}]][i - n], {i, 0, 10}, {n, 0, i}] // Flatten

G.f. of column k: Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(j^k).

G.f. of column k: exp(Sum_{j>=1} sigma_(k+1)(j)*x^j/(j*(1 - x^j))).

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A288389 Expansion of Product_{k>=1} (1 - x^k)^(sigma_2(k)).

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A301556 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma_2(k)).

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A321876 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - x^j)^sigma_k(j).

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