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.

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

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

A288392 Expansion of Product_{k>=1} (1 - x^k)^(sigma_3(k)).

Original entry on oeis.org

1, -1, -9, -19, -9, 163, 573, 1127, 109, -7198, -27159, -58611, -50378, 157532, 892986, 2431694, 4040909, 1605559, -16109148, -68261139, -167737209, -263590908, -109589779, 934422499, 3976197701, 9922490735, 16765911071, 13022553978, -33008232762
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2017

Keywords

Links

Crossrefs

Cf. A027848, A288391.
Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), A288385 (m=1), A288389 (m=2), this sequence (m=3).

Programs

  • Magma
    m:=30; 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):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[3](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..30);  # Alois P. Heinz, Jun 08 2017
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 30 2018 *)
  • PARI
    m=30; x='x+O('x^m); Vec(prod(k=1, m, (1-x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

Formula

Convolution inverse of A288391.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^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)))

A316366 Expansion of Product_{k>=1} (1 - x^k)^A000593(k).

Original entry on oeis.org

1, -1, -1, -3, 3, -1, 5, -1, 19, -12, 7, -33, 34, -64, 22, -106, 150, -148, 203, -230, 534, -411, 626, -806, 1385, -1566, 1503, -2850, 3313, -4691, 4430, -7459, 9799, -11676, 13897, -18173, 27345, -28847, 38230, -47015, 68659, -76681, 94256, -124803, 161835, -200269
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2018

Keywords

Links

Crossrefs

Cf. A000593, A288385, A301799, A301800.

Formula

Convolution inverse of A301799.
Showing 1-5 of 5 results.

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