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 1-9 of 9 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)

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

A363628 Expansion of Sum_{k>0} (1/(1-x^k)^3 - 1).

Original entry on oeis.org

3, 9, 13, 24, 24, 47, 39, 69, 68, 96, 81, 153, 108, 165, 170, 222, 174, 292, 213, 342, 302, 363, 303, 523, 375, 492, 474, 615, 468, 766, 531, 783, 686, 810, 726, 1101, 744, 999, 938, 1248, 906, 1402, 993, 1413, 1306, 1437, 1179, 1901, 1314, 1773, 1562, 1938, 1488, 2238, 1698
Offset: 1

Views

Author

Seiichi Manyama, Jun 12 2023

Keywords

Crossrefs

Cf. A007503, A116963.
Cf. A007437, A069153, A363610.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + 2, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+2, 2));

Formula

G.f.: Sum_{k>0} binomial(k+2,2) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+2,2).

A363640 Expansion of Sum_{k>0} (1/(1 - k*x^k)^4 - 1).

Original entry on oeis.org

4, 18, 32, 91, 76, 358, 148, 917, 796, 2368, 408, 10354, 612, 16586, 16984, 52873, 1208, 180408, 1616, 374934, 271408, 749070, 2692, 3350370, 178376, 4592968, 4349008, 13197802, 5076, 45402484, 6108, 74470417, 64515400, 149432876, 10324768, 652324677, 10028
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A338662, A362683, A363639.
Cf. A116963, A363648.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^# * Binomial[# + 3, 3] &]; Array[a, 40] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^d*binomial(d+3, 3));

Formula

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

A366985 a(n) = Sum_{k=1..n} binomial(k+3,3) * floor(n/k).

Original entry on oeis.org

4, 18, 42, 91, 151, 269, 393, 607, 851, 1207, 1575, 2183, 2747, 3561, 4457, 5640, 6784, 8452, 9996, 12158, 14326, 17004, 19608, 23306, 26642, 30870, 35174, 40518, 45482, 52214, 58202, 65930, 73458, 82382, 90998, 102295, 112179, 124393, 136457, 151125, 164373
Offset: 1

Views

Author

Seiichi Manyama, Oct 30 2023

Keywords

Crossrefs

Partial sums of A116963.
Cf. A006218, A366983, A366984.
Cf. A366971.

Programs

  • PARI
    a(n) = sum(k=1, n, binomial(k+3, 3)*(n\k));
  • Python
    from math import isqrt
def A366985(n): return (-(s:=isqrt(n))*(s*(s*(s*(s+11)+45)+85)+74)+sum(((q:=n//w)+1)*(q*(q*(q+9)+26)+((w+4)*(w*(w+2)+3)<<2)) for w in range(1,s+1)))//3>>3 # Chai Wah Wu, Oct 31 2023

Formula

G.f.: 1/(1-x) * Sum_{k>0} (1/(1-x^k)^4 - 1) = 1/(1-x) * Sum_{k>0} binomial(k+3,3) * x^k/(1-x^k).

A363660 a(n) = Sum_{d|n} binomial(d+n,n).

Original entry on oeis.org

2, 9, 24, 90, 258, 1043, 3440, 13419, 48850, 187836, 705444, 2725099, 10400614, 40233015, 155133856, 601820876, 2333606238, 9079958260, 35345263820, 137876637843, 538259060526, 2104292500739, 8233430727624, 32248866496625, 126410606580284
Offset: 1

Views

Author

Seiichi Manyama, Jun 14 2023

Keywords

Crossrefs

Cf. A007503, A116963, A363628.
Cf. A000984, A134758, A343548, A343549, A363661.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + n, n] &]; Array[a, 25] (* Amiram Eldar, Jul 17 2023 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+n, n));

Formula

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

A363695 Expansion of Sum_{k>0} (1/(1-x^k)^5 - 1).

Original entry on oeis.org

5, 20, 40, 90, 131, 265, 335, 585, 755, 1147, 1370, 2155, 2385, 3410, 4042, 5430, 5990, 8295, 8860, 11843, 13020, 16335, 17555, 23125, 23882, 29805, 32220, 39440, 40925, 51644, 52365, 64335, 67450, 79820, 82712, 101575, 101275, 120805, 125830, 148089, 149000, 179490
Offset: 1

Views

Author

Seiichi Manyama, Jun 16 2023

Keywords

Links

Crossrefs

Cf. A007503, A116963, A363628, A363696.
Cf. A073570, A363605, A363608.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + 4, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+4, 4));

Formula

G.f.: Sum_{k>0} binomial(k+4,4) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+4,4).

A363696 Expansion of Sum_{k>0} (1/(1-x^k)^6 - 1).

Original entry on oeis.org

6, 27, 62, 153, 258, 545, 798, 1440, 2064, 3282, 4374, 6859, 8574, 12447, 15818, 21789, 26340, 36196, 42510, 56538, 66634, 85125, 98286, 126901, 142764, 178506, 203440, 249909, 278262, 343936, 376998, 457686, 506372, 602118, 659058, 791908, 850674, 1005129, 1094638
Offset: 1

Views

Author

Seiichi Manyama, Jun 16 2023

Keywords

Links

Crossrefs

Cf. A007503, A116963, A363628, A363695.
Cf. A101289, A363606.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + 5, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+5, 5));

Formula

G.f.: Sum_{k>0} binomial(k+5,5) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+5,5).

A116989 a(n) = b(A000579(n+6)) with b(n) = Sum{d|n} d*(d+1)*(d+2)*(d+3)*(d+4)*(d+5)/720.

Original entry on oeis.org

1, 925, 1135716, 593223373, 130220375812, 14195655302684, 893936543319276, 36397263567477054, 1025115791220794876, 21899052879460199956, 372805053916689840596, 5076066733212581886566, 57875038239259949679248
Offset: 0

Views

Author

Jonathan Vos Post, Apr 02 2006

Keywords

Crossrefs

Cf. A000579, A007437, A101289, A116963.

Programs

  • PARI
    a(n) = sumdiv(binomial(n+6, 6), d, d*(d+1)*(d+2)*(d+3)*(d+4)*(d+5)/720) /* Georg Fischer, Aug 03 2025 */

Extensions

Definition corrected by Georg Fischer, Aug 03 2025
Showing 1-9 of 9 results.

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