A069153 -id:A069153 - cp's OEIS frontend

A363022 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^3.

0, 1, -3, 7, -10, 13, -21, 35, -39, 36, -55, 85, -78, 71, -118, 155, -136, 130, -171, 232, -234, 177, -253, 389, -310, 248, -390, 455, -406, 378, -465, 651, -586, 426, -626, 832, -666, 533, -822, 1040, -820, 734, -903, 1129, -1144, 783, -1081, 1637, -1197, 961, -1414, 1580, -1378

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A325940, A363598, A363613, A363614.

Cf. A002129, A069153, A321543.

a[n_] := DivisorSum[n, (-1)^# * Binomial[#, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=60, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^3)))

a(n) = sumdiv(n, d, (-1)^d*binomial(d, 2));

G.f.: Sum_{k>0} binomial(k,2) * (-x)^k/(1 - x^k).

a(n) = Sum_{d|n} (-1)^d * binomial(d,2) = (A002129(n) - A321543(n))/2.

A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

Original entry on oeis.org

0, 1, 4, 11, 20, 40, 56, 95, 124, 186, 220, 336, 364, 512, 584, 775, 816, 1129, 1140, 1526, 1600, 1992, 2024, 2720, 2620, 3290, 3400, 4176, 4060, 5280, 4960, 6231, 6208, 7362, 7216, 9195, 8436, 10280, 10248, 12270, 11480, 14432, 13244, 16192, 15884, 18240

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A000292, A032741, A065608, A069153, A363605, A363606.
Cf. A059358, A363607, A363611.
Cf. A000203, A001158, A013662, A092348.

a[n_] := (DivisorSigma[3, n] - DivisorSigma[1, n])/6; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^4)))

a(n) = my(f = factor(n)); (sigma(f, 3) - sigma(f))/6; \\ Amiram Eldar, Dec 30 2024

a(n) = (sigma_3(n) - sigma(n))/6 = A092348(n)/6.
G.f.: Sum_{k>0} binomial(k+1,3) * x^k/(1 - x^k).
From Amiram Eldar, Dec 30 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-3) - zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A363607 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^4.

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 35, 60, 85, 130, 165, 245, 286, 399, 466, 620, 680, 921, 969, 1274, 1366, 1705, 1771, 2325, 2310, 2886, 3010, 3679, 3654, 4666, 4495, 5580, 5622, 6664, 6590, 8285, 7770, 9405, 9426, 11210, 10660, 13230, 12341, 14953, 14740, 16951, 16215, 20181

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013662, A069153, A363608.
Cf. A059358, A363604, A363611.
Cf. A000203, A001157, A001158.

a[n_] := DivisorSum[n, Binomial[#, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^4)))

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

G.f.: Sum_{k>0} binomial(k,3) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d,3).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_3(n) - 3*sigma_2(n) + 2*sigma_1(n)) / 6.
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)

A363605 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^5.

Original entry on oeis.org

0, 1, 5, 16, 35, 76, 126, 226, 335, 531, 715, 1092, 1365, 1947, 2420, 3286, 3876, 5251, 5985, 7861, 8986, 11342, 12650, 16252, 17585, 21841, 24086, 29367, 31465, 38946, 40920, 49662, 53080, 62782, 66206, 80082, 82251, 97376, 102640, 120001, 123410, 146628

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013663, A032741, A065608, A069153, A363604, A363606.

Cf. A000203, A001157, A001158, A001159.

a[n_] := DivisorSum[n, Binomial[# + 2, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^5)))

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

G.f.: Sum_{k>0} binomial(k+2,4) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+2,4).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_4(n) + 2*sigma_3(n) - sigma_2(n) - 2*sigma_1(n)) / 24.
Dirichlet g.f.: zeta(s) * (zeta(s-4) + 2*zeta(s-3) - zeta(s-2) - 2*zeta(s-1)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

A363606 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

Original entry on oeis.org

0, 1, 6, 22, 56, 133, 252, 484, 798, 1344, 2002, 3157, 4368, 6441, 8630, 12112, 15504, 21274, 26334, 35014, 42762, 55133, 65780, 84349, 98336, 123124, 143304, 176373, 201376, 247380, 278256, 336744, 379000, 451402, 502250, 600055, 658008, 775733, 855042

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013664, A032741, A065608, A069153, A363604, A363605.

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

a[n_] := DivisorSum[n, Binomial[# + 3, 5] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^6)))

a(n) = my(f = factor(n)); (sigma(f, 5) + 5*sigma(f, 4) + 5*sigma(f, 3) - 5*sigma(f, 2) - 6*sigma(f)) / 120; \\ Amiram Eldar, Dec 30 2024

G.f.: Sum_{k>0} binomial(k+3,5) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+3,5).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_5(n) + 5*sigma_4(n) + 5*sigma_3(n) - 5*sigma_2(n) - 6*sigma_1(n)) / 120.
Dirichlet g.f.: zeta(s) * (zeta(s-5) + 5*zeta(s-4) + 5*zeta(s-3) - 5*zeta(s-2) - 6*zeta(s-1)) / 120.
Sum_{k=1..n} a(k) ~ (zeta(6)/720) * n^6. (End)

A363610 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^3.

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 15, 24, 29, 42, 45, 69, 66, 93, 98, 129, 120, 175, 153, 216, 206, 255, 231, 343, 282, 366, 354, 447, 378, 550, 435, 594, 542, 648, 582, 828, 630, 819, 770, 978, 780, 1114, 861, 1161, 1072, 1221, 1035, 1529, 1143, 1494, 1346, 1644, 1326, 1878, 1482, 1953

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A000005, A000203, A001157, A002117, A065608, A363611.

Cf. A007437, A069153.

a[n_] := DivisorSum[n, Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^3)))

a(n) = my(f = factor(n)); (sigma(f, 2) - 3*sigma(f) + 2*numdiv(f)) / 2; \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{k>0} binomial(k-1,2) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d-1,2).
From Amiram Eldar, Jan 01 2025: (Start)
a(n) = (sigma_2(n) - 3*sigma_1(n) + 2*sigma_0(n)) / 2.
Dirichlet g.f.: zeta(s) * (zeta(s-2) - 3*zeta(s-1) + 2*zeta(s)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)

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

Seiichi Manyama, Jun 12 2023

Cf. A007503, A116963.

Cf. A007437, A069153, A363610.

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));
```

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

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

A363608 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^5.

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 71, 126, 215, 330, 511, 715, 1036, 1370, 1891, 2380, 3201, 3876, 5061, 6020, 7645, 8855, 11207, 12655, 15665, 17676, 21512, 23751, 29000, 31465, 37851, 41250, 48756, 52400, 62602, 66045, 77691, 82966, 96521, 101270, 118966, 123410, 143397

Offset: 1

Seiichi Manyama, Jun 11 2023

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

Cf. A013663, A069153, A363607.

Cf. A000203, A001157, A001158, A001159.

a[n_] := DivisorSum[n, Binomial[#, 4] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^k)^5)))

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

G.f.: Sum_{k>0} binomial(k,4) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d,4).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_4(n) - 6*sigma_3(n) + 11*sigma_2(n) - 6*sigma_1(n)) / 24.
Dirichlet g.f.: zeta(s) * (zeta(s-4) - 6*zeta(s-3) + 11*zeta(s-2) - 6*zeta(s-1)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

A086666 a(n) = sigma_2(n) - sigma_1(n).

Original entry on oeis.org

0, 2, 6, 14, 20, 38, 42, 70, 78, 112, 110, 182, 156, 226, 236, 310, 272, 416, 342, 504, 468, 574, 506, 790, 620, 808, 780, 994, 812, 1228, 930, 1302, 1172, 1396, 1252, 1820, 1332, 1750, 1644, 2120, 1640, 2404, 1806, 2478, 2288, 2578, 2162, 3286, 2394, 3162

Offset: 1

Jon Perry, Jul 27 2003

Total area of all distinct L X W rectangles such that s + t = n, 1 <= s <= t, s | n, L = n/s and W = t/s. - Wesley Ivan Hurt, Aug 01 2025

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000203, A001157, A002117, A052847, A069153, A092348.

Table[DivisorSigma[2,n]-DivisorSigma[1,n],{n,50}] (* Harvey P. Dale, Aug 01 2020 *)

PARI
```
a(n) = sigma(n,2)-sigma(n,1);
```

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

G.f.: Sum_{n>=1} n*(n-1) * x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k-1)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
From Peter Bala, Jan 21 2021: (Start)
a(n) = 2*A069153(n).
G.f.: A(x) = Sum_{n >= 1} 2*x^(2*n)/(1 - x^n)^3.
A faster converging series: A(x) = 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)).
Sum_{k=1..n} a(k) ~ (zeta(3)/3) * n^3. (End)
a(n) = Sum_{d|n} d*(d-1). - Wesley Ivan Hurt, Aug 01 2025

A366967 a(n) = Sum_{k=2..n} binomial(k,2) * floor(n/k).

Original entry on oeis.org

0, 1, 4, 11, 21, 40, 61, 96, 135, 191, 246, 337, 415, 528, 646, 801, 937, 1145, 1316, 1568, 1802, 2089, 2342, 2737, 3047, 3451, 3841, 4338, 4744, 5358, 5823, 6474, 7060, 7758, 8384, 9294, 9960, 10835, 11657, 12717, 13537, 14739, 15642, 16881, 18025, 19314, 20395

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A069153.
Cf. A002541, A236632.
Cf. A024916, A064602, A366971.

a(n) = sum(k=2, n, binomial(k, 2)*(n\k));

from math import isqrt
def A366967(n): return ((s:=isqrt(n))**2*(1-s**2)+sum((q:=n//k)*(3*k*(k-1)+q**2-1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 30 2023

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

a(n) = (A064602(n)-A024916(n))/2. - Chai Wah Wu, Oct 30 2023

cp's OEIS Frontend

A363022 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^3.

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A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

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A363607 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^4.

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A363605 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^5.

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A363606 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^6.

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A363610 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^3.

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

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A363608 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^5.

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