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.

A343547 a(n) = n * Sum_{d|n} binomial(d+n-2,n-1)/d.

Original entry on oeis.org

1, 4, 9, 32, 75, 318, 931, 3712, 13014, 50110, 184767, 715656, 2704169, 10454976, 40126395, 155462016, 601080407, 2335849578, 9075135319, 35359120940, 137847221148, 538346579034, 2104098963743, 8234009441952, 32247603785500, 126414311404108, 495918587420145
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A309731, A332508, A343544, A343545, A343546.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + n - 2, n-1]/# &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+n-2, n-1)/d);

Formula

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

A343544 a(n) = n * Sum_{d|n} binomial(d+2,3)/d.

Original entry on oeis.org

1, 6, 13, 32, 40, 94, 91, 184, 204, 320, 297, 612, 468, 770, 850, 1184, 986, 1752, 1349, 2280, 2114, 2662, 2323, 4184, 3125, 4264, 4266, 5740, 4524, 7660, 5487, 8352, 7546, 9180, 8470, 13212, 9176, 12654, 12194, 16640, 12382, 19628, 14233, 20724, 19590, 22034, 18471, 30416, 21462
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A059358, A309731, A343545, A343546, A343547.

Programs

  • Maple
    f:= n -> n/6*add((d+1)*(d+2),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Apr 26 2021
  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 2, 3]/# &]; Array[a, 50] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+2, 3)/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+2, 3)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^4 = Sum_{k>=1} binomial(k+2,3) * x^k/(1 - x^k)^2.

A343549 a(n) = n * Sum_{d|n} binomial(d+n-1,n)/d.

Original entry on oeis.org

1, 5, 13, 49, 131, 545, 1723, 6809, 24484, 94445, 352727, 1366273, 5200313, 20135939, 77571083, 301034537, 1166803127, 4540794476, 17672631919, 68943346009, 269129827042, 1052178506615, 4116715363823, 16124644677569, 63205303337656, 247964681424725, 973469783435197
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A309731, A343544, A343545, A343546, A343547, A343548.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + n - 1, n]/# &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+n-1, n)/d);

Formula

a(n) = [x^n] Sum_{k>=1} k * x^k/(1 - x^k)^(n+1).
a(n) = [x^n] Sum_{k>=1} binomial(k+n-1,n) * x^k/(1 - x^k)^2.
From Seiichi Manyama, Jun 14 2023: (Start)
a(n) = Sum_{d|n} binomial(d+n-1,d).
a(n) = [x^n] Sum_{k>=1} (1/(1 - x^k)^n - 1). (End)

A343546 a(n) = n * Sum_{d|n} binomial(d+4,5)/d.

Original entry on oeis.org

1, 8, 24, 72, 131, 318, 469, 936, 1359, 2294, 3014, 5172, 6201, 9548, 12126, 17376, 20366, 29862, 33668, 47372, 54684, 71874, 80753, 111000, 119410, 154986, 173988, 220864, 237365, 309864, 324663, 411744, 445170, 542776, 578984, 731340, 749435, 918118, 981474
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Crossrefs

Cf. A000203, A038040, A101289, A309731, A343544, A343545, A343547.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 4, 5]/# &]; Array[a, 40] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+4, 5)/d);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+4, 5)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^6 = Sum_{k>=1} binomial(k+4,5) * x^k/(1 - x^k)^2.

A366934 Expansion of Sum_{k>=1} k^5 * x^k/(1 - x^k)^5.

Original entry on oeis.org

1, 37, 258, 1219, 3195, 9597, 17017, 39338, 63189, 118580, 162052, 316974, 373113, 630959, 826320, 1262692, 1424702, 2353896, 2483414, 3912790, 4397862, 6003569, 6451293, 10240908, 10004850, 13819832, 15382332, 20810398, 20547109, 30847530, 28675527, 40458504, 41853306
Offset: 1

Views

Author

Seiichi Manyama, Oct 29 2023

Keywords

Crossrefs

Cf. A064987, A366135, A366933.
Cf. A073570, A343545.
Cf. A343573.

Programs

  • PARI
    a(n) = sumdiv(n, d, d^5*binomial(n/d+3, 4));

Formula

a(n) = Sum_{d|n} d^5 * binomial(n/d+3,4).
Showing 1-5 of 5 results.

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