A007437 -id:A007437 - cp's OEIS frontend

A301873 Expansion of Product_{k>=1} 1/(1 - x^k)^A007437(k).

1, 1, 5, 12, 36, 80, 215, 476, 1154, 2539, 5772, 12417, 27146, 57111, 120822, 249389, 514201, 1041684, 2103211, 4189502, 8306632, 16296337, 31803839, 61530913, 118413823, 226200319, 429857982, 811633548, 1524828119, 2848379512, 5295550209

Offset: 0

Vaclav Kotesovec, Mar 28 2018

nonn

Euler transform of A007437.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A007437, A301874.

nmax = 40; CoefficientList[Series[Exp[Sum[Sum[(DivisorSigma[1, k] + DivisorSigma[2, k]) * x^(j*k) / (2*j), {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

a(n) ~ exp(2^(7/4) * Pi * Zeta(3)^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) + sqrt(5*Zeta(3)*n/6)/2 - (7*Pi * 5^(1/4) / (2^(15/4) * 3^(7/4) * Zeta(3)^(1/4)) + 5^(5/4) * Zeta(3)^(3/4) / (2^(15/4) * 3^(3/4) * Pi)) * n^(1/4) + (17*Zeta(3))/(72*Pi^2) + 23/576) * A^(1/4) * Zeta(3)^(23/192) / (2^(307/192) * 15^(23/192) * n^(119/192)), where A is the Glaisher-Kinkelin constant A074962.

A301874 Expansion of Product_{k>=1} (1 + x^k)^A007437(k).

Original entry on oeis.org

1, 1, 4, 11, 27, 64, 156, 345, 779, 1706, 3665, 7742, 16207, 33300, 67830, 136526, 271969, 536588, 1049801, 2035620, 3917547, 7482738, 14192358, 26738962, 50062081, 93158467, 172366532, 317166618, 580542738, 1057269629, 1916174666

Offset: 0

Vaclav Kotesovec, Mar 28 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A007437, A301873.

nmax = 40; CoefficientList[Series[Exp[Sum[-(-1)^j * Sum[(DivisorSigma[1, k] + DivisorSigma[2, k]) * x^(j*k) / (2*j), {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

a(n) ~ exp(2*Pi * (7*Zeta(3))^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) + sqrt(15*Zeta(3)*n/7)/4 - (5^(1/4) * 7^(3/4) * Pi / (3^(7/4) * Zeta(3)^(1/4)) + 15^(5/4) * Zeta(3)^(3/4) / (7^(5/4)*Pi)) * n^(1/4)/16 + 75*Zeta(3) / (784*Pi^2) + 5/192) * (7*Zeta(3))^(1/8) / (2^(95/48) * 15^(1/8) * n^(5/8)).

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

Original entry on oeis.org

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)

A069153 a(n) = Sum_{d|n} d*(d-1)/2.

Original entry on oeis.org

0, 1, 3, 7, 10, 19, 21, 35, 39, 56, 55, 91, 78, 113, 118, 155, 136, 208, 171, 252, 234, 287, 253, 395, 310, 404, 390, 497, 406, 614, 465, 651, 586, 698, 626, 910, 666, 875, 822, 1060, 820, 1202, 903, 1239, 1144, 1289, 1081, 1643, 1197, 1581, 1414, 1736

Offset: 1

Benoit Cloitre, Apr 08 2002

Inverse Mobius transform of A000217. - R. J. Mathar, Jan 19 2009

			x^2 + 3*x^3 + 7*x^4 + 10*x^5 + 19*x^6 + 21*x^7 + 35*x^8 + 39*x^9 + 56*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000203, A001157, A002117, A007437, A086666, A134840.

Cf. A032741, A065608, A363604, A363605, A363606.

with(numtheory):
seq((1/2)*(sigma[2](n) - sigma[1](n)), n = 1..100); # Peter Bala, Jan 21 2021

A069153[n_]:=Plus@@Binomial[Divisors[n],2];Array[A069153,100] (* Enrique Pérez Herrero, Feb 21 2012 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 - d) / 2)}

a(n) = my(f = factor(n)); (sigma(f, 2) - sigma(f)) / 2; \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{k>0} x^(2*k)/(1-x^k)^3. - Vladeta Jovovic, Dec 17 2002
Row sums of triangle A134840. - Gary W. Adamson, Nov 12 2007
G.f. A(x) = (1/2) * x * d/dx log( B(x) ) where B() is g.f. for A052847. - Michael Somos, Feb 12 2008
G.f.: Sum_{k>0} ((k^2 - k) / 2) * x^k / (1 - x^k). - Michael Somos, Feb 12 2008
From Peter Bala, Jan 21 2021: (Start)
a(n) = (1/2)*(sigma_2(n) - sigma_1(n)) = (1/2)*(A001157(n)  A000203(n)) =  (1/2)*A086666.
G.f.: A(x) = (1/2)* Sum_{n >= 1} x^(n^2)*( n*(n-1)*x^(3*n) - (n^2 + n - 2)*x^(2*n) + n*(3 - n)*x^n + n*(n - 1) )/(1 - x^n)^3. - differentiate equation 5 in Arndt twice w.r.t x and set x = 1. (End)
From Amiram Eldar, Jan 01 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-2) - zeta(s-1)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (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

A116963 Inverse Moebius transform of the shifted tetrahedral numbers.

Original entry on oeis.org

4, 14, 24, 49, 60, 118, 124, 214, 244, 356, 368, 608, 564, 814, 896, 1183, 1144, 1668, 1544, 2162, 2168, 2678, 2604, 3698, 3336, 4228, 4304, 5344, 4964, 6732, 5988, 7728, 7528, 8924, 8616, 11297, 9884, 12214, 12064, 14668, 13248, 17132, 15184, 18928, 18412, 21038

Offset: 1

Jonathan Vos Post, Mar 31 2006

			a(12) = ((1+1)*(1+2)*(1+3)/6) + ((2+1)*(2+2)*(2+3)/6) + ((3+1)*(3+2)*(3+3)/6) + ((4+1)*(4+2)*(4+3)/6) + ((6+1)*(6+2)*(6+3)/6) + ((12+1)*(12+2)*(12+3)/6) = 4 + 10 + 20 + 35 + 84 + 455 = 608.
a(13) = ((1+1)*(1+2)*(1+3)/6) + ((13+1)*(13+2)*(13+3)/6) = 4 + 560 = 564.

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

See also: A007437 (inverse Moebius transform of triangular numbers).
Cf. A000292, A007437, A007503.
Cf. A013662, A059358, A363604, A363607, A363611.
Cf. A000005, A000203, A001157, A001158.

a[n_] := DivisorSum[n, Binomial[# + 3, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 05 2023 *)

my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, 1/(1-x^k)^4-1)) \\ Seiichi Manyama, Jun 12 2023

a(n) = Sum_{d|n} (d+1)*(d+2)*(d+3)/6 = Sum_{d|n} A000292(d+1).
G.f.: Sum_{k>0} (1/(1-x^k)^4 - 1). - Seiichi Manyama, Jun 12 2023
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_3(n) + 6*sigma_2(n) + 11*sigma_1(n) + 6*sigma_0(n))/6.
Dirichlet g.f.: zeta(s) * (zeta(s-3) + 6*zeta(s-2) + 11*zeta(s-1) + 6*zeta(s)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

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

Jonathan Vos Post, Mar 31 2006

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)

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

Cf. A000389, A007437, A013664, A073570, A116963, A116969, A332508.

Cf. A000203, A001157, A001158, A001159, A001160.

a(n) = sumdiv(n, d, binomial(d+4, 5)); \\ Seiichi Manyama, Apr 19 2021

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

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

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)

A309731 Expansion of Sum_{k>=1} k * x^k/(1 - x^k)^3.

Original entry on oeis.org

1, 5, 9, 20, 20, 48, 35, 76, 72, 110, 77, 204, 104, 196, 210, 288, 170, 405, 209, 480, 378, 440, 299, 816, 425, 598, 594, 868, 464, 1200, 527, 1104, 858, 986, 910, 1800, 740, 1216, 1170, 1960, 902, 2184, 989, 1980, 1890, 1748, 1175, 3216, 1470, 2475, 1938, 2704, 1484, 3456, 2090

Offset: 1

Ilya Gutkovskiy, Aug 14 2019

nonn

Dirichlet convolution of natural numbers (A000027) with triangular numbers (A000217).

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

Cf. A000005, A000027, A000203, A000217, A007437, A007503, A034715, A038040, A064987, A309732.

with(numtheory): seq(n*(tau(n)+sigma(n))/2, n=1..30); # Ridouane Oudra, Nov 28 2019

nmax = 55; CoefficientList[Series[Sum[k x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j, j (j + 1)/2, j, n], {n, 1, 55}]
Table[n (DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 55}]

a(n)=sumdiv(n,d,binomial(n/d+1,2)*d); \\ Andrew Howroyd, Aug 14 2019

a(n)=n*(numdiv(n) + sigma(n))/2; \\ Andrew Howroyd, Aug 14 2019

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+1, 2)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Apr 19 2021

G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k/(1 - x^k)^2.
a(n) = n * (d(n) + sigma(n))/2.
Dirichlet g.f.: zeta(s-1) * (zeta(s-2) + zeta(s-1))/2.
a(n) = Sum_{k=1..n} k*tau(gcd(n,k)). - Ridouane Oudra, Nov 28 2019

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).

cp's OEIS Frontend

A301873 Expansion of Product_{k>=1} 1/(1 - x^k)^A007437(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A301874 Expansion of Product_{k>=1} (1 + x^k)^A007437(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

A069153 a(n) = Sum_{d|n} d*(d-1)/2.

Views

Author

Keywords

Comments

Examples

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

A116963 Inverse Moebius transform of the shifted tetrahedral numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A101289 Inverse Moebius transform of 5-simplex numbers A000389.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI