A101289 -id:A101289 - cp's OEIS frontend

A059358 Coefficients in expansion of Sum_{n >= 1} x^n/(1-x^n)^4.

0, 1, 5, 11, 25, 36, 71, 85, 145, 176, 260, 287, 455, 456, 649, 726, 961, 970, 1376, 1331, 1820, 1866, 2315, 2301, 3175, 2961, 3736, 3830, 4729, 4496, 5966, 5457, 6945, 6842, 8114, 7890, 10196, 9140, 11215, 11126, 13420, 12342, 15730, 14191, 17515, 17106, 19601

Offset: 0

N. J. A. Sloane, Jan 27 2001

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000203, A000292, A001157, A001158, A002129, A007437, A013662, A048272, A073570, A101289.

a:= proc(n) option remember;
      add(d*(d+1)*(d+2)/6, d=numtheory[divisors](n))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 12 2023

With[{nn=50},CoefficientList[Series[Sum[x^n/(1-x^n)^4,{n,nn}],{x,0,nn}],x]] (* Harvey P. Dale, May 14 2013 *)

a(n) = if(n==0, 0, sumdiv(n, d, binomial(d+2, 3))); \\ Seiichi Manyama, Apr 19 2021

a(n) = if(n==0, 0, my(f = factor(n)); (sigma(f, 3) + 3*sigma(f, 2) + 2 * sigma(f)) / 6); \\ Amiram Eldar, Dec 29 2024

a(n) = (1/6)*(sigma_3(n) + 3*sigma_2(n) + 2*sigma_1(n)), i.e., this sequence is the inverse Möbius transform of tetrahedral (or pyramidal) numbers: n*(n+1)(n+2)/6 with g.f. 1/(1-x)^4 (cf. A000292). - Vladeta Jovovic, Aug 31 2002
L.g.f.: -log(Product_{k>=1} (1 - x^k)^((k+1)*(k+2)/6)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
From Amiram Eldar, Dec 29 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-3) + 3*zeta(s-2) + 2*zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A073570 G.f.: Sum_{n >= 1} x^n/(1-x^n)^5.

Original entry on oeis.org

1, 6, 16, 41, 71, 147, 211, 371, 511, 791, 1002, 1547, 1821, 2596, 3146, 4247, 4846, 6627, 7316, 9681, 10852, 13657, 14951, 19427, 20546, 25577, 27916, 34096, 35961, 44912, 46377, 56607, 59922, 70896, 74096, 90278, 91391, 108591, 113766, 133421

Offset: 1

Vladeta Jovovic, Aug 31 2002

nonn

Inverse Moebius transform of pentatope numbers. - Jonathan Vos Post, Mar 31 2006

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A000332, A007437, A013663, A059358, A101289.

Cf. A000203, A001157, A001158, A001159.

Table[(DivisorSigma[4,n]+6*DivisorSigma[3,n]+11*DivisorSigma[2,n]+ 6*DivisorSigma[ 1,n])/24,{n,40}] (* Harvey P. Dale, Aug 08 2013 *)

a(n) = sumdiv(n, d, binomial(d+3, 4)); \\ Seiichi Manyama, Apr 19 2021

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+3, 4)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021

a(n) = my(f = factor(n)); (sigma(f, 4) + 6*sigma(f, 3) + 11*sigma(f, 2) + 6*sigma(f)) / 24; \\ Amiram Eldar, Dec 30 2024

a(n) = (1/24) * (sigma_4(n) + 6*sigma_3(n) + 11*sigma_2(n) + 6*sigma_1(n)).
a(n) = Sum_{d|n} (d+1)*(d+2)*(d+3)*(d+4)/24 = Sum_{d|n} C(d+3,4) = Sum_{d|n} A000332(d+3). - Jonathan Vos Post, Mar 31 2006. Corrected by Joshua Zucker, May 04 2007
From Amiram Eldar, Dec 30 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-4) + 6*zeta(s-3) + 11*zeta(s-2) + 6*zeta(s-2)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007

A332508 a(n) = Sum_{d|n} binomial(n+d-2, n-1).

Original entry on oeis.org

1, 3, 7, 25, 71, 280, 925, 3561, 12916, 49346, 184757, 710255, 2704157, 10427747, 40119781, 155288897, 601080391, 2334714319, 9075135301, 35352181116, 137846759282, 538302226628, 2104098963721, 8233718962365, 32247603703576, 126412458920775, 495918551104687

Offset: 1

Ilya Gutkovskiy, Feb 14 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A007437, A059358, A073570, A101289, A308814, A332470.

Table[DivisorSum[n, Binomial[n + # - 2, n - 1] &], {n, 1, 27}]
Table[SeriesCoefficient[Sum[x^k/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 1, 27}]

a(n) = sumdiv(n, d, binomial(n+d-2, n-1)); \\ Michel Marcus, Feb 14 2020

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

a(n) ~ 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 04 2022

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

Seiichi Manyama, Apr 19 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

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

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

a(n) = sumdiv(n, d, binomial(d+n-1, n));

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).

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

Seiichi Manyama, Apr 19 2021

nonn

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

a[n_] := n * DivisorSum[n, Binomial[# + 4, 5]/# &]; Array[a, 40] (* Amiram Eldar, Apr 25 2021 *)

a(n) = n*sumdiv(n, d, binomial(d+4, 5)/d);

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+4, 5)*x^k/(1-x^k)^2))

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.

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

Seiichi Manyama, Jun 16 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007503, A116963, A363628, A363695.

Cf. A101289, A363606.

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));
```

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

Jonathan Vos Post, Apr 02 2006

Cf. A000579, A007437, A101289, A116963.

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 */

Definition corrected by Georg Fischer, Aug 03 2025

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

Seiichi Manyama, Oct 31 2023

			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, ...

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

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

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

cp's OEIS Frontend

A059358 Coefficients in expansion of Sum_{n >= 1} x^n/(1-x^n)^4.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A073570 G.f.: Sum_{n >= 1} x^n/(1-x^n)^5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A332508 a(n) = Sum_{d|n} binomial(n+d-2, n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Crossrefs

Programs

PARI

Extensions

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).

Views

Author

Keywords

Examples

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.