cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A288007 Expansion of 1/Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

Original entry on oeis.org

1, -1, -1, -1, 1, 1, 0, 2, 2, -1, -2, 0, -1, -1, -4, -1, 2, 0, -1, 2, 2, 5, -1, 4, 8, -4, -5, 0, -1, -1, -6, -1, 3, -7, -9, -5, 1, 3, -3, 3, 17, 0, -6, 8, 12, 8, 0, 8, 17, -11, -9, -10, 0, -2, -20, 5, 14, -18, -25, -10, 1, -7, -21, 2, 29, -12, -17, 6, 17, 32, -4
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2017

Keywords

Links

Crossrefs

Cf. A107742, A109386.
Product_{k>=1} 1/(1 + x^k)^sigma_m(k): this sequence (m=0), A288421 (m=1), A288422 (m=2), A288423 (m=3).

Programs

  • Magma
    m:=80; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(&*[(&*[1 + x^(j*k): j in [1..2*m]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018
  • Maple
    with(numtheory): seq(coeff(series(exp(-add(sigma(k)*x^k/(k*(1-x^(2*k))),k=1..n)),x,n+1), x, n), n = 0 .. 70); # Muniru A Asiru, Jan 30 2019
  • Mathematica
    A109386[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2] &] &]; a[0] = 1; a[n_] := a[n] = -(1/n) Sum[A109386[k] a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 04 2017 *)
CoefficientList[Series[1/Product[Product[1+x^(j*k), {j,1,100}], {k,1,100}], {x,0,80}], x] (* G. C. Greubel, Oct 29 2018 *)
  • PARI
    m=80; x='x+O('x^m); Vec(1/(prod(k=1,2*m, prod(j=1,2*m, 1+x^(j*k) )))) \\ G. C. Greubel, Oct 29 2018

Formula

Convolution inverse of A107742.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 26 2018

A288422 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_2(k)).

Original entry on oeis.org

1, -1, -4, -6, 0, 24, 51, 89, 47, -152, -578, -1149, -1482, -738, 2384, 8901, 18476, 26774, 24151, -7143, -86804, -226605, -406442, -539872, -441822, 181268, 1671148, 4240334, 7618777, 10551340, 10218856, 1973258, -20190349, -61492391, -121880826
Offset: 0

Views

Author

Seiichi Manyama, Jun 09 2017

Keywords

Links

Crossrefs

Cf. A288414, A288419.
Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), A288421 (m=1), this sequence (m=2), A288423 (m=3).

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(2,k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)
  • PARI
    m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1+x^k)^sigma(k,2))) \\ G. C. Greubel, Oct 29 2018

Formula

Convolution inverse of A288414.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288419(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018

A288423 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_3(k)).

Original entry on oeis.org

1, -1, -8, -20, -8, 134, 512, 1062, 406, -5319, -22532, -51843, -58869, 83035, 648412, 1947384, 3665081, 3040131, -8272126, -46481039, -128400098, -234847560, -215189896, 378947363, 2437661943, 7036096665, 13868464378, 16886982518, -4042283985, -93095770772
Offset: 0

Views

Author

Seiichi Manyama, Jun 09 2017

Keywords

Links

Crossrefs

Cf. A288415, A288420.
Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), A288421 (m=1), A288422 (m=2), this sequence (m=3).

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(3,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
  • Maple
    with(numtheory): seq(coeff(series(mul(1/(1+x^k)^(sigma[3](k)),k=1..n),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 31 2018
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)
  • PARI
    m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1+x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

Formula

Convolution inverse of A288415.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288420(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Oct 25 2018

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^k)/(1+x^k))^sigma(k)))
Showing 1-4 of 4 results.

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