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 1-8 of 8 results.

A324158 Expansion of Sum_{k>=1} x^k / (1 - k * x^k)^k.

Original entry on oeis.org

1, 2, 2, 6, 2, 23, 2, 50, 56, 107, 2, 660, 2, 499, 1592, 2370, 2, 8246, 2, 18557, 21786, 11387, 2, 175198, 43752, 53419, 298892, 487762, 2, 1891098, 2, 2552066, 3905222, 1114403, 3785462, 29081597, 2, 4981099, 48376512, 95510772, 2, 218764940, 2, 346411232, 770590352
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 02 2019

Keywords

Links

Crossrefs

Cf. A055225, A087909, A157019, A157020, A167531, A324159.

Programs

  • Mathematica
    nmax = 45; CoefficientList[Series[Sum[x^k/(1 - k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(n/d)^(d - 1) Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}], {n, 1, 45}]
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d+d-2, d-1)); \\ Michel Marcus, Sep 02 2019
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, x^k/(1-k*x^k)^k)) \\ Seiichi Manyama, Sep 03 2019

Formula

a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(n/d+d-2,d-1).
a(p) = 2, where p is prime.

A157020 a(n) = Sum_{d|n} d*binomial(n/d+d-2,d-1).

Original entry on oeis.org

1, 3, 4, 9, 6, 22, 8, 33, 28, 46, 12, 131, 14, 78, 136, 177, 18, 307, 20, 456, 302, 166, 24, 1149, 376, 222, 568, 1177, 30, 2387, 32, 1761, 958, 358, 2556, 5224, 38, 438, 1496, 7851, 42, 8317, 44, 4863, 9136, 622, 48, 20169, 6518, 11451, 3112, 8516, 54, 23734
Offset: 1

Views

Author

R. J. Mathar, Feb 21 2009

Keywords

Comments

Equals row sums of triangle A157497. [Gary W. Adamson & Mats Granvik, Mar 01 2009]

Links

Crossrefs

Cf. A081543, A132065, A156833 (Mobius transform), A324158, A324159.

Programs

  • Maple
    add( d*binomial(n/d+d-2,d-1),d=numtheory[divisors](n) ) ;
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, k*x^k/(1-x^k)^k)) \\ Seiichi Manyama, Sep 03 2019

Formula

G.f.: Sum_{n>=1} n*x^n/(1-x^n)^n.

A338661 a(n) = Sum_{d|n} d^n * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 289, 3126, 49036, 823544, 17040385, 387538588, 10048833246, 285311670612, 8929334253419, 302875106592254, 11116754387182648, 437894348359764856, 18448995959423107073, 827240261886336764178, 39347761059781438793815, 1978419655660313589123980
Offset: 1

Views

Author

Seiichi Manyama, Apr 22 2021

Keywords

Links

Crossrefs

Cf. A023887, A157019, A157020, A324158, A324159, A339481, A339482, A339712, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^n * Binomial[# + n/# - 2, #-1] &]; Array[a, 20] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^n*binomial(d+n/d-2, d-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-(k*x)^k))^k))

Formula

G.f.: Sum_{k >= 1} (k * x/(1 - (k * x)^k))^k.
If p is prime, a(p) = 1 + p^p.
a(n) ~ n^n. - Vaclav Kotesovec, Aug 04 2025

A343573 a(n) = Sum_{d|n} d^d * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 265, 3126, 46750, 823544, 16778257, 387420652, 10000015646, 285311670612, 8916100731047, 302875106592254, 11112006831322846, 437893890380906656, 18446744073843774497, 827240261886336764178, 39346408075300025340205
Offset: 1

Views

Author

Seiichi Manyama, Apr 20 2021

Keywords

Links

Crossrefs

Cf. A023887, A157019, A157020, A324158, A324159, A338661, A339481, A339482, A339712, A343567, A343574.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^#*Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 20 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^d*binomial(d+n/d-2, d-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-x^k))^k))

Formula

G.f.: Sum_{k >= 1} (k * x/(1 - x^k))^k.
If p is prime, a(p) = 1 + p^p.

A339481 a(n) = Sum_{d|n} d^(n-d) * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 2, 2, 10, 2, 131, 2, 1282, 4376, 16907, 2, 1138272, 2, 5793475, 154455992, 469893122, 2, 49501130330, 2, 1318441711177, 19001093813466, 3138439911059, 2, 15989399214596398, 6675720214843752, 3937376603803099, 6754271297694102092, 47097064577536888014, 2
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A157019, A157020, A324158, A324159, A338661, A339482, A339712, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n - #) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(n-d)*binomial(d+n/d-2, d-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (x/(1-(k*x)^k))^k))

Formula

G.f.: Sum_{k >= 1} (x/(1 - (k * x)^k))^k.
If p is prime, a(p) = 2.

A339712 a(n) = Sum_{d|n} d^(d+n/d-1) * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 273, 3126, 46948, 823544, 16781441, 387421948, 10000078446, 285311670612, 8916102176891, 302875106592254, 11112006865913416, 437893890382064056, 18446744074783625217, 827240261886336764178, 39346408075327954053967, 1978419655660313589123980
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A157019, A157020, A324158, A324159, A338661, A339481, A339482, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/# - 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d-1)*binomial(d+n/d-2, d-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-k*x^k))^k))

Formula

G.f.: Sum_{k >= 1} (k * x/(1 - k * x^k))^k.
If p is prime, a(p) = 1 + p^p.

A339482 a(n) = Sum_{d|n} d^(n-d+1) * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 3, 4, 21, 6, 346, 8, 4617, 13132, 80696, 12, 4903847, 14, 40410966, 756336736, 2416181265, 18, 306560794753, 20, 6941876836216, 132964265599502, 34522735212626, 24, 116720277621236637, 33378601074218776, 51185893450298400, 60788365423272068968
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A157019, A157020, A324158, A324159, A338661, A339481, A339712, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n - # + 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(n-d+1)*binomial(d+n/d-2, d-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k*(x/(1-(k*x)^k))^k))

Formula

G.f.: Sum_{k >= 1} k * (x/(1 - (k * x)^k))^k.
If p is prime, a(p) = 1 + p.

A338688 a(n) = - Sum_{d|n} (-n/d)^d * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 1, 4, -5, 6, 2, 8, -121, 172, 44, 12, -759, 14, 566, 5536, -7665, 18, -6877, 20, 2744, 70862, 21218, 24, -570573, 218776, 104324, 918568, 942479, 30, -3693495, 32, -9408481, 11779582, 2223344, 19935756, -15628120, 38, 9954650, 145283360, -371959011, 42, -382916059
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A217670, A324159, A338683, A338689.

Programs

  • Mathematica
    a[n_] := -DivisorSum[n, (-n/#)^# * Binomial[# + n/# - 2, # - 1] &]; Array[a, 40] (* Amiram Eldar, Apr 24 2021 *)
  • PARI
    a(n) = -sumdiv(n, d, (-n/d)^d*binomial(d+n/d-2, d-1));
  • PARI
    N=66; x='x+O('x^N); Vec(sum(k=1, N, k*(x/(1+k*x^k))^k))

Formula

G.f.: Sum_{k>=1} k * (x/(1 + k * x^k))^k.
If p is prime, a(p) = (-1)^(p-1) + p.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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