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 1-6 of 6 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.

A117108 Moebius transform of tetrahedral numbers.

Original entry on oeis.org

1, 3, 9, 16, 34, 43, 83, 100, 155, 182, 285, 292, 454, 473, 636, 696, 968, 929, 1329, 1304, 1678, 1735, 2299, 2136, 2890, 2818, 3489, 3484, 4494, 4052, 5455, 5168, 6250, 6168, 7652, 6988, 9138, 8547, 10196, 9840, 12340, 10954, 14189, 13140, 15380, 14993, 18423
Offset: 1

Views

Author

Steve Butler, Apr 18 2006

Keywords

Comments

Partial sums of a(n) give A015634(n).
See also A059358, A116963 (applied to shifted version of tetrahedral numbers), inverse Moebius transform of tetrahedral numbers. - Jonathan Vos Post, Apr 20 2006

Examples

			a(2) = 3 because of the triples (1,1,1), (1,1,2), (1,2,2).

Links

Crossrefs

Cf. A000292 (tetrahedral numbers), A007438, A008683, A015634 (partial sums), A059358, A116963, A117109, A343544.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#]*Binomial[# + 2, 3] &]; Array[a, 50] (* Amiram Eldar, Jun 07 2025 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+2, 3)*moebius(n/d)); \\ Michel Marcus, Nov 04 2018

Formula

a(n) = |{(x,y,z) : 1 <= x <= y <= z <= n, gcd(x,y,z,n) = 1}|.
G.f.: Sum_{k>=1} mu(k) * x^k / (1 - x^k)^4. - Ilya Gutkovskiy, Feb 13 2020

Extensions

Offset changed to 1 by Ilya Gutkovskiy, Feb 13 2020

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)

A343545 a(n) = n * Sum_{d|n} binomial(d+3,4)/d.

Original entry on oeis.org

1, 7, 18, 49, 75, 177, 217, 428, 549, 890, 1012, 1824, 1833, 2849, 3360, 4732, 4862, 7506, 7334, 10810, 11382, 14729, 14973, 22188, 20850, 27482, 29052, 37408, 35989, 50490, 46407, 61824, 62106, 75854, 75390, 101673, 91427, 116033, 117624, 146680, 135792, 179886, 163228, 208208
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Crossrefs

Cf. A000203, A038040, A073570, A309731, A343544, A343546, A343547.

Programs

  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 3, 4]/# &]; Array[a, 50] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+3, 4)/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+3, 4)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^5 = Sum_{k>=1} binomial(k+3,4) * x^k/(1 - x^k)^2.

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.

A366933 Expansion of Sum_{k>=1} k^4 * x^k/(1 - x^k)^4.

Original entry on oeis.org

1, 20, 91, 340, 660, 1836, 2485, 5560, 7536, 13280, 14927, 31360, 29016, 49924, 60390, 89776, 84490, 152496, 131651, 226520, 227066, 299420, 282141, 514080, 415425, 581776, 614070, 850864, 711776, 1226520, 928977, 1442400, 1362042, 1693064, 1644930, 2609076
Offset: 1

Views

Author

Seiichi Manyama, Oct 29 2023

Keywords

Crossrefs

Cf. A064987, A366135, A366934.
Cf. A059358, A343544.
Cf. A343573.

Programs

  • PARI
    a(n) = sumdiv(n, d, d^4*binomial(n/d+2, 3));

Formula

a(n) = Sum_{d|n} d^4 * binomial(n/d+2,3).
Showing 1-6 of 6 results.

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