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.

A318784 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

Original entry on oeis.org

1, 0, 1, 1, 4, 2, 11, 6, 25, 20, 56, 44, 139, 107, 283, 266, 619, 567, 1317, 1242, 2680, 2705, 5403, 5539, 10947, 11339, 21291, 23013, 41494, 45213, 79991, 88312, 151546, 170908, 284901, 324421, 532505, 611227, 981002, 1142000, 1797451, 2105773, 3268765, 3855050, 5889704, 7004451
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 03 2018

Keywords

Comments

Convolution of A061256 and A073592.
Euler transform of A001065.

Links

Crossrefs

Cf. A000203, A001065, A001157, A061256, A073592, A318783.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      (sigma(d)-d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 03 2018
  • Mathematica
    nmax = 45; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^(2 k)/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A001065(k).
G.f.: exp(Sum_{k>=1} sigma_2(k)*x^(2*k)/(k*(1 - x^k))), where sigma_2(k) = sum of squares of divisors of k (A001157).
a(n) ~ exp(3^(2/3) * c^(1/3) * n^(2/3)/2 - Pi^2 * n^(1/3) / (4 * 3^(2/3) * c^(1/3)) - Pi^4/(288*c) - 1/8) * A^(3/2) * c^(1/8) / (3^(5/8) * (2*Pi)^(11/24) * n^(5/8)), where c = (Pi^2 - 6)*Zeta(3) and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Sep 03 2018

A321260 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(sigma_n(k)-k^n).

Original entry on oeis.org

1, 0, 1, 1, 18, 2, 861, 132, 106024, 40910, 72980055, 6838271, 228282942581, 27620223647, 2050169324675668, 352809815149813, 87174966874755673105, 6798293425492905407, 18318448554980083512011863, 1187839217207171380193247, 11258918803635775614062752424535
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Crossrefs

Cf. A283333, A318783, A318784, A319647, A321258, A321261.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^k)^(DivisorSigma[n, k] - k^n), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^(2 k)/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^(2*k)/(k*(1 - x^k))).

A318844 Expansion of Product_{k>=1} (1 + x^k)^(d(k)-1), where d(k) = number of divisors of k (A000005).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 4, 8, 10, 15, 17, 29, 31, 48, 60, 81, 99, 143, 167, 231, 287, 374, 460, 615, 740, 964, 1194, 1512, 1856, 2379, 2877, 3635, 4460, 5540, 6759, 8433, 10192, 12608, 15335, 18774, 22726, 27868, 33525, 40863, 49292, 59652, 71694, 86780, 103818, 125118, 149778, 179608
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 04 2018

Keywords

Comments

Convolution of A081362 and A107742.
Weigh transform of A032741.

Links

Crossrefs

Cf. A000005, A000203, A032741, A081362, A107742, A318783.

Programs

  • Maple
    with(numtheory): a:=series(mul((1+x^k)^(tau(k)-1),k=1..100),x=0,53): seq(coeff(a,x,n),n=0..52); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 52; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 52; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, k] - 1) x^k/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (DivisorSigma[0, d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A032741(k).
G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/(k*(1 - x^(2*k)))), where sigma_1(k) = sum of divisors of k (A000203).

A327798 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*(j + 1))).

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 9, 10, 25, 34, 72, 106, 215, 330, 635, 1025, 1899, 3141, 5713, 9602, 17213, 29292, 51982, 89149, 157249, 271027, 476037, 823386, 1442063, 2500015, 4370386, 7588146, 13248591, 23026728, 40169991, 69865026, 121811765, 211954826, 369412910
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2019

Keywords

Comments

Invert transform of A032741.

Links

Crossrefs

Cf. A032741, A129921, A318783, A327739.

Programs

  • Maple
    N:= 100: # for a(0)..a(N)
G:= 1/(1-add(x^(2*k)/(1-x^k),k=1..(N+1)/2)):
S:= series(G,x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Jan 10 2023
  • Mathematica
    nmax = 38; CoefficientList[Series[1/(1 - Sum[x^(2 k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(DivisorSigma[0, k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 38}]

Formula

G.f.: 1 / (1 - Sum_{k>=1} x^(2*k) / (1 - x^k)).
a(0) = 1; a(n) = Sum_{k=1..n} A032741(k) * a(n-k).
Showing 1-4 of 4 results.

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