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.

A101289 Inverse Moebius transform of 5-simplex numbers A000389.

Original entry on oeis.org

1, 7, 22, 63, 127, 280, 463, 855, 1309, 2135, 3004, 4704, 6189, 9037, 11776, 16359, 20350, 27901, 33650, 44695, 53614, 68790, 80731, 103776, 118882, 148701, 171220, 210469, 237337, 292292, 324633, 393351, 438922, 522298, 576346, 690333, 749399
Offset: 1

Views

Author

Jonathan Vos Post, Mar 31 2006

Keywords

Comments

From Georg Fischer, Aug 06 2025: (Start)
The general pattern is a(n) = Sum_{d|n} (Product_{k=0..m-1} d+k)/m! = Sum_{d|n} binomial(d+m-1, m) = Sum{d|n} Axxxxxx(d), with:
m Axxxxxx resulting sequence
------------------------------
2 A000217 A007437
3 A000292 A116963
4 A000332 A073570
5 A000389 A101289 (this sequence)
The other formulas generalize correspondingly.
A116989 uses A000579 and m=6 within a modified formula.
(End)

Links

Crossrefs

Cf. A000389, A007437, A013664, A073570, A116963, A116969, A332508.
Cf. A000203, A001157, A001158, A001159, A001160.

Programs

  • PARI
    a(n) = sumdiv(n, d, binomial(d+4, 5)); \\ Seiichi Manyama, Apr 19 2021
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+4, 5)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021
  • PARI
    a(n) = my(f = factor(n));  (sigma(f, 5) + 10*sigma(f, 4) + 35*sigma(f, 3) + 50*sigma(f, 2) + 24*sigma(f))/120; \\ Amiram Eldar, Dec 30 2024

Formula

a(n) = Sum_{d|n} d*(d+1)*(d+2)*(d+3)*(d+4)/120 = Sum_{d|n} C(d+4,5) = Sum{d|n} A000389(d) = Sum_{d|n} (d^5+10*d^4+35*d^3+50*d^2+24*d)/120.
G.f.: Sum_{k>=1} x^k/(1 - x^k)^6 = Sum_{k>=1} binomial(k+4,5) * x^k/(1 - x^k). - Seiichi Manyama, Apr 19 2021
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_5(n) + 10*sigma_4(n) + 35*sigma_3(n) + 50*sigma_2(n) + 24*sigma_1(n)) / 120.
Dirichlet g.f.: zeta(s) * (zeta(s-5) + 10*zeta(s-4) + 35*zeta(s-3) + 50*zeta(s-2) + 24*zeta(s-1)) / 120.
Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

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.

A332470 a(n) = Sum_{d|n} mu(n/d) * binomial(n+d-2, n-1).

Original entry on oeis.org

1, 1, 5, 16, 69, 226, 923, 3312, 12825, 47896, 184755, 700712, 2704155, 10373455, 40113421, 154946976, 601080389, 2332498482, 9075135299, 35338355380, 137846298360, 538213522254, 2104098963719, 8233142596640, 32247603662625, 126408753954731, 495918514791900
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 13 2020

Keywords

Links

Crossrefs

Cf. A000010, A007438, A008683, A117108, A117109, A332508, A343547.

Programs

  • Magma
    [&+[MoebiusMu(n div d) *Binomial(n+d-2,n-1):d in Divisors(n)]:n in [1..30]]; // Marius A. Burtea, Feb 13 2020
  • Mathematica
    Table[DivisorSum[n, MoebiusMu[n/#] Binomial[n + # - 2, n - 1] &], {n, 1, 27}]
Table[SeriesCoefficient[Sum[MoebiusMu[k] x^k/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 1, 27}]
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*binomial(n+d-2, n-1)); \\ Michel Marcus, Feb 14 2020

Formula

a(n) = [x^n] Sum_{k>=1} mu(k) * x^k / (1 - x^k)^n.
a(n) = |{(x_1, x_2, ... , x_{n-1}) : 1 <= x_1 <= x_2 <= ... <= x_n = n, gcd(x_1, x_2, ... , x_n) = 1}|. - Seiichi Manyama, Apr 20 2021

A343548 a(n) = Sum_{d|n} binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 41, 127, 498, 1717, 6610, 24366, 93391, 352717, 1358826, 5200301, 20097076, 77562773, 300786339, 1166803111, 4539163784, 17672631901, 68933291834, 269129233484, 1052113994124, 4116715363801, 16124221819056, 63205303242628, 247961973949228, 973469736360283
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000005, A000203, A007437, A059358, A073570, A101289, A332508, A343549.

Programs

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

Formula

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

A343517 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n <= n} gcd(x_1, x_2, ... , x_n, n).

Original entry on oeis.org

1, 4, 12, 42, 130, 506, 1722, 6622, 24426, 93427, 352726, 1359388, 5200312, 20097156, 77567064, 300787366, 1166803126, 4539197723, 17672631918, 68933307843, 269129530770, 1052113994340, 4116715363822, 16124224571368, 63205303313900, 247961973949536
Offset: 1

Views

Author

Seiichi Manyama, Apr 17 2021

Keywords

Links

Crossrefs

Main diagonal of A343516.
Cf. A000010, A156834, A332508, A332517.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, EulerPhi[n/#] * Binomial[n + # - 1, n] &]; Array[a, 25] (* Amiram Eldar, Apr 18 2021 *)
  • PARI
    a(n) = sumdiv(n, d, eulerphi(n/d)*binomial(d+n-1, n));

Formula

a(n) = Sum_{d|n} phi(n/d) * binomial(d+n-1, n).
a(n) = [x^n] Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^(n+1).
a(n) ~ 2^(2*n - 1) / sqrt(Pi*n). - Vaclav Kotesovec, May 23 2021

A343553 a(n) = Sum_{1 <= x_1 <= x_2 <= ... <= x_n = n} gcd(x_1, x_2, ... , x_n).

Original entry on oeis.org

1, 3, 8, 26, 74, 287, 930, 3572, 12966, 49379, 184766, 710712, 2704168, 10427822, 40123208, 155289768, 601080406, 2334740919, 9075135318, 35352194658, 137846990678, 538302226835, 2104098963742, 8233721100024, 32247603765020, 126412458921072, 495918569262798
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Examples

			a(3) = gcd(1,1,3) + gcd(1,2,3) + gcd(1,3,3) + gcd(2,2,3) + gcd(2,3,3) + gcd(3,3,3) = 1 + 1 + 1 + 1 + 1 + 3 = 8.

Links

Crossrefs

Cf. A000010, A332470, A332508, A343516, A343517, A343547.

Programs

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

Formula

a(n) = A343516(n,n-1).
a(n) = Sum_{d|n} phi(n/d) * binomial(d+n-2, n-1).
a(n) = [x^n] Sum_{k >= 1} phi(k) * x^k/(1 - x^k)^n.
a(n) ~ 2^(2*n - 2) / sqrt(Pi*n). - Vaclav Kotesovec, May 23 2021

A363666 a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+n-2,n-1).

Original entry on oeis.org

1, 3, 7, 29, 71, 355, 925, 4425, 13276, 60111, 184757, 856357, 2704157, 12137147, 40367461, 176999505, 601080391, 2616894901, 9075135301, 38884056181, 138014377810, 583674491643, 2104098963721, 8823912454489, 32247616479976, 133998376789707
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A167531, A363642, A363645.
Cf. A332508, A363663, A363667.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + n - 2, n - 1] &]; Array[a, 25] (* Amiram Eldar, Jul 12 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+n-2, n-1));

Formula

a(n) = [x^n] Sum_{k>0} x^k/(1 - k*x^k)^n.

A366986 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} binomial(d+k-1,k).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 4, 3, 1, 5, 7, 7, 2, 1, 6, 11, 14, 6, 4, 1, 7, 16, 25, 16, 12, 2, 1, 8, 22, 41, 36, 31, 8, 4, 1, 9, 29, 63, 71, 71, 29, 15, 3, 1, 10, 37, 92, 127, 147, 85, 50, 13, 4, 1, 11, 46, 129, 211, 280, 211, 145, 52, 18, 2, 1, 12, 56, 175, 331, 498, 463, 371, 176, 74, 12, 6
Offset: 1

Views

Author

Seiichi Manyama, Oct 31 2023

Keywords

Examples

			Square  array begins:
  1,  1,  1,  1,   1,   1,   1, ...
  2,  3,  4,  5,   6,   7,   8, ...
  2,  4,  7, 11,  16,  22,  29, ...
  3,  7, 14, 25,  41,  63,  92, ...
  2,  6, 16, 36,  71, 127, 211, ...
  4, 12, 31, 71, 147, 280, 498, ...
  2,  8, 29, 85, 211, 463, 925, ...

Crossrefs

Columns k=0..5 give A000005, A000203, A007437, A059358, A073570, A101289.
T(n,n-1) gives A332508.
T(n,n) gives A343548.
Cf. A366977.

Programs

  • PARI
    T(n, k) = sumdiv(n, d, binomial(d+k-1, k));

Formula

G.f. of column k: Sum_{j>=1} x^j/(1 - x^j)^(k+1).
Showing 1-8 of 8 results.

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

Content is available under The OEIS End-User License Agreement.