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 1-5 of 5 results.

A288391 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_3(k)).

Original entry on oeis.org

1, 1, 10, 38, 156, 534, 2014, 6796, 23312, 76165, 247234, 780343, 2435903, 7453859, 22538336, 67130594, 197666509, 574876417, 1654464954, 4711217687, 13288453688, 37133349758, 102873771662, 282630567325, 770410193747, 2084205092693, 5598070811010
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2017

Keywords

Links

Crossrefs

Cf. A027848, A288392, A288415.
Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (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):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[3](d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 08 2017
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
  • 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

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.
a(n) ~ exp((5*Pi)^(4/5) * Zeta(5)^(1/5) * n^(4/5) / (2^(8/5) * 3^(1/5)) - Zeta'(-3)/2) * Zeta(5)^(121/1200) / ((24*Pi)^(121/1200) * 5^(721/1200) * n^(721/1200)). - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A288098 Convolution inverse of A006171.

Original entry on oeis.org

1, -1, -2, 0, 0, 4, 1, 3, 0, -5, 0, -7, -6, -4, 7, 0, 6, 9, 11, 10, -2, 13, -13, -10, -17, -20, -25, 0, -11, -11, -2, 11, 41, 27, 41, 17, 58, 12, 27, -21, -2, -36, -67, -52, -59, -95, -75, -20, -89, 35, 0, 62, 41, 142, 97, 172, 63, 154, 148, 85, 110, -36, -17, -156
Offset: 0

Views

Author

Seiichi Manyama, Jun 05 2017

Keywords

Links

Crossrefs

Cf. A006171, A060640, A107742.
Product_{k>=1} (1 - x^k)^sigma_m(k): this sequence (m=0), A288385 (m=1), A288389 (m=2), A288392 (m=3).

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 - x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Aug 28 2018 *)
nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Aug 28 2018 *)

Formula

G.f.: Product_{n>=1} E(q^n) where E(q) = Product_{n>=1} (1-q^n).
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A060640(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Aug 26 2018

A288415 Expansion of Product_{k>=1} (1 + x^k)^(sigma_3(k)).

Original entry on oeis.org

1, 1, 9, 37, 137, 487, 1749, 5901, 19695, 63832, 202905, 632689, 1941394, 5860868, 17448558, 51255292, 148726841, 426605755, 1210569740, 3400427281, 9460683203, 26083933370, 71300381025, 193313191005, 520057831035, 1388722752205, 3682100198763
Offset: 0

Views

Author

Seiichi Manyama, Jun 09 2017

Keywords

Links

Crossrefs

Cf. A288391, A288392, A288420.
Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), A192065 (m=1), A288414 (m=2), this sequence (m=3).

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+q^k)^DivisorSigma(3,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
  • Maple
    with(numtheory): seq(coeff(series(mul((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+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+x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

Formula

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288420(k)*a(n-k) for n > 0.
a(n) ~ exp(5*Pi^(4/5) * Zeta(5)^(1/5) * n^(4/5) / 2^(12/5)) * Zeta(5)^(1/10) / (2^(169/240) * sqrt(5) * Pi^(1/10) * n^(3/5)). - Vaclav Kotesovec, Mar 23 2018
G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^3). - Ilya Gutkovskiy, Aug 26 2018

A288385 Expansion of Product_{k>=1} (1 - x^k)^sigma(k).

Original entry on oeis.org

1, -1, -3, -1, 0, 10, 8, 12, 1, -28, -29, -67, -51, -28, 79, 163, 256, 343, 273, 136, -351, -649, -1446, -1751, -1889, -1453, -124, 1924, 5138, 7608, 10636, 10903, 10054, 3143, -5799, -20521, -37217, -53057, -65661, -66086, -54430, -15648, 37179, 122732
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2017

Keywords

Links

Crossrefs

Cf. A001001, A061256.
Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), this sequence (m=1), A288389 (m=2), A288392 (m=3).

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma(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..45);  # Alois P. Heinz, Jun 08 2017
  • Mathematica
    b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*DivisorSigma[1, d], {d,
     Divisors[j]}]*b[n - j], {j, 1, n}]/n];
a[n_] := a[n] = If[n == 0, 1, -Sum[b[n - i]*a[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

Formula

Convolution inverse of A061256.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A001001(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 29 2018

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jun 08 2017

Keywords

Links

Crossrefs

Cf. A027847.
Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), A288385 (m=1), this sequence (m=2), A288392 (m=3).

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-q^k)^DivisorSigma(2,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
  • Maple
    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
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 30 2018 *)
  • PARI
    m=50; x='x+O('x^m); Vec(prod(k=1, m, (1-x^k)^sigma(k,2))) \\ G. C. Greubel, Oct 30 2018

Formula

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
Showing 1-5 of 5 results.

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