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 1-10 of 10 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.

A324159 Expansion of Sum_{k>=1} k * x^k / (1 - k * x^k)^k.

Original entry on oeis.org

1, 3, 4, 13, 6, 58, 8, 137, 172, 296, 12, 2063, 14, 1254, 5536, 7697, 18, 25201, 20, 68976, 70862, 23882, 24, 607485, 218776, 108720, 918568, 1810089, 30, 6746147, 32, 9408545, 11779582, 2233172, 19935756, 102405280, 38, 9968370, 145283360, 393585971, 42, 730233631, 44, 1296043651, 2718300016
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 02 2019

Keywords

Links

Crossrefs

Cf. A055225, A087909, A157019, A157020, A324158.

Programs

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

Formula

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

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.

A156834 A156348 * A000010.

Original entry on oeis.org

1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11, 63, 13, 56, 99, 89, 17, 154, 19, 269, 237, 132, 23, 509, 301, 182, 379, 783, 29, 1230, 31, 881, 813, 306, 2125, 2431, 37, 380, 1299, 4157, 41, 4822, 43, 3695, 6175, 552, 47, 8529, 5587, 6266, 2787
Offset: 1

Views

Author

Gary W. Adamson, Feb 16 2009

Keywords

Comments

Conjecture: for n>1, a(n) = n iff n is prime. Companion to A156833.

Examples

			a(4) = 5 = (1, 2, 0, 1) dot (1, 1, 2, 2) = (1 + 2 + 0 + 2), where row 4 of A156348 = (1, 2, 0, 1) and (1, 1, 2, 2) = the first 4 terms of Euler's phi function.

Links

Crossrefs

Cf. A156348, A000010, A156833, A157020, A343553.
Equals row sums of triangle A157030. [Gary W. Adamson, Feb 21 2009]

Programs

  • Maple
    A156834 := proc(n)
        add(A156348(n,k)*numtheory[phi](k),k=1..n) ;
end proc: # R. J. Mathar, Mar 03 2013
  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[# + n/# - 2, #-1] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-2, d-1)); \\ Seiichi Manyama, Apr 22 2021
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*(x/(1-x^k))^k)) \\ Seiichi Manyama, Apr 22 2021

Formula

Equals A156348 * A054525 * [1, 2, 3,...]; where A054525 = the inverse Mobius transform.
a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-2, d-1). - Seiichi Manyama, Apr 22 2021
G.f.: Sum_{k >= 1} phi(k) * (x/(1 - x^k))^k. - Seiichi Manyama, Apr 22 2021

Extensions

Extended beyond a(14) by R. J. Mathar, Mar 03 2013

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.

A157497 Triangle read by rows, A156348 * A127648.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 4, 0, 4, 1, 0, 0, 0, 5, 1, 6, 9, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 8, 0, 16, 0, 0, 0, 8, 1, 0, 18, 0, 0, 0, 0, 0, 9, 1, 10, 0, 0, 25, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 12, 30, 40, 0, 36, 0, 0, 0, 0, 0, 12
Offset: 1

Views

Author

Gary W. Adamson & Mats Granvik, Mar 01 2009

Keywords

Comments

Row sums = A157020: (1, 3, 4, 9, 6, 22, 8,...)

Examples

			First few rows of the triangle =
1;
1, 2;
1, 0, 3;
1, 4, 0, 4;
1, 0, 0, 0, 5;
1, 6, 9, 0, 0, 6;
1, 0, 0, 0, 0, 0, 7;
1, 8, 0, 16, 0, 0, 0, 8;
1, 0, 18, 0, 0, 0, 0, 0, 9;
1, 10, 0, 0, 25, 0, 0, 0, 0, 10;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11;
1, 12, 30, 40, 0, 36, 0, 0, 0, 0, 0, 12;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
1, 14, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 14;
...
Row 4 = (1, 4, 0, 4) = termwise products of (1, 2, 0, 1) and (1, 2, 3, 4)
where (1, 2, 0, 1) = row 4 of triangle A156348.

Crossrefs

Cf. A156348, A126648, A156020

Formula

Triangle read by rows, A156348 * A127648. A127648 = an infinite lower triangular matrix with (1, 2, 3,...) as the main diagonal and the rest zeros.

A338656 a(n) = Sum_{d|n} mu(d) * binomial(d + n/d - 2, d-1).

Original entry on oeis.org

1, 0, 0, -1, 0, -4, 0, -3, -5, -8, 0, -9, 0, -12, -28, -7, 0, -8, 0, -34, -54, -20, 0, 9, -69, -24, -44, -83, 0, 0, 0, -15, -130, -32, -418, 157, 0, -36, -180, -129, 0, 0, 0, -285, -494, -44, 0, 633, -923, -24, -304, -454, 0, 1090, -2000, -1183, -378, -56, 0, 3050, 0, -60, -3002, -31, -3638, 0
Offset: 1

Views

Author

Seiichi Manyama, Apr 22 2021

Keywords

Links

Crossrefs

Cf. A008683, A157020, A332470, A338657.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[#] * Binomial[# + n/# - 2, # - 1] &]; Array[a, 100] (* Amiram Eldar, Apr 22 2021 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(d)*binomial(d+n/d-2, d-1));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*(x/(1-x^k))^k))

Formula

G.f.: Sum_{k >= 1} mu(k) * (x/(1 - x^k))^k.
If p is prime, a(p) = 0.
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