A015128 -id:A015128 - cp's OEIS frontend

A002132 Generalized sum of divisors function.

1, 2, 4, 8, 14, 18, 28, 40, 52, 70, 88, 104, 140, 168, 196, 240, 278, 320, 380, 440, 504, 562, 644, 720, 808, 910, 1000, 1120, 1240, 1360, 1488, 1600, 1789, 1938, 2100, 2296, 2452, 2660, 2880, 3080, 3292, 3542, 3784, 4048, 4400, 4572, 4868, 5280, 5502, 5850

Offset: 4

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 4..10000
G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

A diagonal of A060047.

Cf. A015128.

nmax = 60; Drop[CoefficientList[Series[1/2 * Sum[(-1)^k*k*Binomial[k + 1, 3]*x^(k^2), {k, 2, nmax}]/(1 + 2*Sum[(-x)^(k^2), {k, 1, nmax}]), {x, 0, nmax}], x], 4] (* Vaclav Kotesovec, Jul 30 2025 *)

G.f.: (1/2) * ( Sum_{k>=2} (-1)^k * k * binomial(k+1,3) * q^(k^2) ) / ( 1 + 2 * Sum_{k>=1} (-q)^(k^2) ). - Seiichi Manyama, Sep 15 2023

More terms from Naohiro Nomoto and Vladeta Jovovic, Jan 25 2002

A060046 Generalized sum of divisors function: third diagonal of A060047.

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 40, 56, 84, 122, 168, 232, 312, 408, 528, 672, 865, 1078, 1336, 1648, 2002, 2424, 2912, 3472, 4116, 4872, 5744, 6648, 7752, 8976, 10304, 11872, 13566, 15424, 17556, 19896, 22414, 25256, 28336, 31584, 35462, 39482, 43728, 48664

Offset: 9

N. J. A. Sloane, Mar 19 2001

Seiichi Manyama, Table of n, a(n) for n = 9..10000
G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, arXiv:1010.5769 [math.NT], 2010.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Cf. A015128.

nmax = 60; Drop[CoefficientList[Series[-1/3 * Sum[(-1)^k*k*Binomial[k + 2, 5]*x^(k^2), {k, 3, nmax}]/(1 + 2*Sum[(-x)^(k^2), {k, 1, nmax}]), {x, 0, nmax}], x], 9] (* Vaclav Kotesovec, Jul 30 2025 *)

G.f.: (t(1)^3-3*t(1)*t(2)+2*t(3))/6 where t(i) = Sum((x^(2*n-1)/(1-x^(2*n-1))^2)^i,n=1..inf), i=1..3. - Vladeta Jovovic, Sep 21 2007

G.f.: -(1/3) * ( Sum_{k>=3} (-1)^k * k * binomial(k+2,5) * q^(k^2) ) / ( 1 + 2 * Sum_{k>=1} (-q)^(k^2) ). - Seiichi Manyama, Sep 15 2023

More terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Sep 21 2007

A261647 Expansion of Product_{k>=0} ((1+x^(2k+1))/(1-x^(2k+1)))^3.

Original entry on oeis.org

1, 6, 18, 44, 102, 216, 428, 816, 1494, 2650, 4584, 7740, 12804, 20808, 33264, 52400, 81462, 125100, 189966, 285516, 425016, 627040, 917436, 1331856, 1919332, 2746926, 3905784, 5519352, 7754064, 10833192, 15055216, 20817600, 28647414, 39241336, 53517060

Offset: 0

Vaclav Kotesovec, Aug 28 2015

nonn

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.

Cf. A015128, A156616.
Cf. A080054, A007096, A014969, A261648, A014970, A014972, A103261.
Cf. A261610, A261649, A261651.
Cf. A261611, A261650, A261652.

nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

a(n) ~ exp(Pi*sqrt(3*n/2)) * 3^(1/4) / (8 * 2^(1/4) * n^(3/4)).

A265844 Expansion of Product_{k>=1} (1 + k^2x^k)/(1 - k^2x^k).

Original entry on oeis.org

1, 2, 10, 36, 118, 376, 1188, 3456, 10054, 28814, 79280, 215844, 581748, 1528456, 3987384, 10295952, 26130982, 65874532, 164661622, 406787220, 998529752, 2434022304, 5879630196, 14124455856, 33734350692, 80000820426, 188787849968, 443372664504, 1035137265552

Offset: 0

Vaclav Kotesovec, Dec 16 2015

nonn

Convolution of A092484 and A077335.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A015128, A077335, A092484, A265758.

nmax = 40; CoefficientList[Series[Product[(1 + k^2*x^k)/(1 - k^2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(2*n/3), where
c = 33024.782174678163138510272317... if mod(n,3) = 0
c = 33024.230416953709449028604542... if mod(n,3) = 1
c = 33024.292470246596667257649964... if mod(n,3) = 2.

A266821 Expansion of Product_{k>=1} (1 + 3*x^k) / (1 - x^k).

Original entry on oeis.org

1, 4, 8, 24, 44, 88, 176, 312, 544, 924, 1584, 2552, 4136, 6488, 10128, 15632, 23748, 35640, 53080, 78136, 114024, 165552, 237744, 339544, 481248, 678236, 949008, 1321840, 1830376, 2521688, 3456672, 4717208, 6406680, 8666448, 11672464, 15660528, 20934868

Offset: 0

Vaclav Kotesovec, Jan 04 2016

nonn

Convolution of A000041 and A032308.

In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 - x^k)) then a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(m+1)*n), where c = 2*Pi^2/3 + log(m)^2 + 2*polylog(2, -1/m).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 5001 terms from Vaclav Kotesovec)

Cf. A000041, A015128, A032308, A264686.

Column k=4 of A321884.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      (t-> b(t, min(t, i-1)))(n-i*j), j=1..n/i)*4 +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..44);  # Alois P. Heinz, Aug 28 2019

nmax = 40; CoefficientList[Series[Product[(1+3*x^k) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

{ my(n=40); Vec(prod(k=1, n, 4/(1-x^k) - 3 + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017

a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (8*Pi*n), where c = 2*Pi^2/3 + log(3)^2 + 2*polylog(2, -1/3) = 7.16861897522987077909937377164783326088308015803... .

A278690 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2 in powers of x.

Original entry on oeis.org

1, 2, 5, 9, 18, 31, 54, 88, 144, 225, 351, 531, 800, 1179, 1728, 2492, 3573, 5058, 7119, 9918, 13743, 18882, 25810, 35028, 47313, 63513, 84883, 112833, 149373, 196803, 258309, 337590, 439650, 570357, 737496, 950270, 1220688, 1563021, 1995642, 2540466, 3225386

Offset: 0

Seiichi Manyama, Nov 26 2016

			G.f. = 1 + 2*x + 5*x^2 + 9*x^3 + 18*x^4 + 31*x^5 + 54*x^6 + ...
G.f. = q + 2*q^25 + 5*q^49 + 9*q^73 + 18*q^97 + 31*q^121 + 54*q^145 + ... - _Michael Somos_, Nov 25 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^k: A000726 (k=1), this sequence (k=2), A273845 (k=3), A182819 (k=4).
Cf. Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^2: A000041 (k=1), A015128 (k=2), this sequence (k=3), A160461 (k=5).
Cf. A298311.

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 26 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 25 2019 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / eta(x + A)^2, n))}; /* Michael Somos, Nov 25 2019 */

G.f.: Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2.
a(n) ~ sqrt(5/3)*exp(sqrt(10*n)*Pi/3)/(12*n). - Vaclav Kotesovec, Nov 26 2016
Expansion of q^(-1/24) * eta(q^3) / eta(q)^2 in powers of q. - Michael Somos, Nov 25 2019
G.f.: 1/Product_{n > = 1} ( 1 - x^(n/gcd(n,k)) ) for k = 3. Cf. A000041 (k = 1), A015128 (k = 2), A298311 (k = 4) and A160461 (k = 5). - Peter Bala, Nov 17 2020

A284592 Square array read by antidiagonals: T(n,k) is the number of pairs of partitions of n and k respectively, such that the pair of partitions have no part in common.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 3, 1, 1, 3, 5, 1, 2, 1, 5, 7, 2, 3, 3, 2, 7, 11, 2, 5, 4, 5, 2, 11, 15, 4, 6, 7, 7, 6, 4, 15, 22, 4, 10, 8, 12, 8, 10, 4, 22, 30, 7, 12, 14, 14, 14, 14, 12, 7, 30, 42, 8, 18, 16, 24, 16, 24, 16, 18, 8, 42, 56, 12, 23, 25, 28, 28, 28, 28, 25, 23, 12, 56

Offset: 0

Peter Bala, Mar 30 2017

Compare with A284593.

			Square array begins
  n\k|  0  1  2  3  4  5   6   7   8   9  10
- - - - - - - - - - - - - - - - - - - - - - -
  0  |  1  1  2  3  5  7  11  15  22  30  42: A000041
  1  |  1  0  1  1  2  2   4   4   7   8  12: A002865
  2  |  2  1  2  3  5  6  10  12  18  23  32
  3  |  3  1  3  4  7  8  14  16  25  31  44
  4  |  5  2  5  7 12 14  24  28  43  54  76
  5  |  7  2  6  8 14 16  28  31  49  60  85
  6  | 11  4 10 14 24 28  48  55  85 106 149
  7  | 15  4 12 16 28 31  55  60  95 115 163
  8  | 22  7 18 25 43 49  85  95 148 182 256
  9  | 30  8 23 31 54 60 106 115 182 220 311
  10 | 42 12 32 44 76 85 149 163 256 311 438
  ...
T(4,3) = 7: the 7 pairs of partitions of 4 and 3 with no parts in common are (4, 3), (4, 2 + 1), (4, 1 + 1 + 1), (2 + 2, 3), (2 + 2, 1 + 1 + 1), (2 + 1 + 1 , 3) and (1 + 1 + 1 + 1, 3).

Alois P. Heinz, Antidiagonals n = 0..200, flattened
H. S. Wilf, Lectures on Integer Partitions

Cf. A000041 (row 0), A002865 (row 1), A015128 (antidiagonal sums), A284593.

Main diagonal gives A054440 or 2*A260669 (for n>0).

#A284592 as a square array
ser := taylor(taylor(mul(1 + x^j/(1 - x^j) + y^j/(1 - y^j), j = 1..10), x, 11), y, 11):
convert(ser, polynom):
s := convert(%, polynom):
with(PolynomialTools):
for n from 0 to 10 do CoefficientList(coeff(s, y, n), x) end do;
# second Maple program:
b:= proc(n, k, i) option remember; `if`(n=0 and
       (k=0 or i=1), 1, `if`(i<1, 0, b(n, k, i-1)+
       add(b(sort([n-i*j, k])[], i-1), j=1..n/i)+
       add(b(sort([n, k-i*j])[], i-1), j=1..k/i)))
    end:
A:= (n, k)-> (l-> b(l[1], l[2]$2))(sort([n, k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Apr 02 2017

Table[Total@ Boole@ Map[! IntersectingQ @@ Map[Union, #] &, Tuples@ {IntegerPartitions@ #, IntegerPartitions@ k}] &[n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 02 2017 *)
b[n_, k_, i_] := b[n, k, i] = If[n == 0 &&
     (k == 0 || i == 1), 1, If[i < 1, 0, b[n, k, i - 1] +
     Sum[b[Sequence @@ Sort[{n - i*j, k}], i - 1], {j, 1, n/i}] +
     Sum[b[Sequence @@ Sort[{n, k - i*j}], i - 1], {j, 1, k/i}]]];
A[n_, k_] := Function [l, b[l[[1]], l[[2]], l[[2]]]][Sort[{n, k}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jun 07 2021, after Alois P. Heinz *)

O.g.f. Product_{j >= 1} (1 + x^j/(1 - x^j) + y^j/(1 - y^j)) = Sum_{n,k >= 0} T(n,k)*x^n*y^k (see Wilf, Example 7).

Antidiagonal sums are A015128.

A285458 Expansion of Product_{k>=1} ((1 + x^k) / (1 - x^(4*k)))^k.

Original entry on oeis.org

1, 1, 2, 5, 9, 17, 30, 54, 94, 161, 269, 449, 740, 1200, 1930, 3083, 4877, 7650, 11919, 18444, 28363, 43341, 65848, 99523, 149654, 223901, 333448, 494427, 729996, 1073408, 1572264, 2294389, 3336191, 4834261, 6981727, 10050944, 14424665, 20639641, 29447118

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

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

Cf. A015128, A000041, A285445, A006950.

Cf. A156616, A000219, A285446, A285459.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^(4*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(1/12 + 3 * (13*Zeta(3))^(1/3) * n^(2/3) / 4) * (13*Zeta(3))^(7/36) / (2 * A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 8, 0, 1, 8, 24, 32, 14, 0, 1, 10, 40, 80, 76, 24, 0, 1, 12, 60, 160, 234, 168, 40, 0, 1, 14, 84, 280, 552, 624, 352, 64, 0, 1, 16, 112, 448, 1110, 1712, 1552, 704, 100, 0, 1, 18, 144, 672, 2004, 3912, 4896, 3648, 1356, 154, 0, 1, 20, 180, 960, 3346, 7896, 12600, 13120, 8184, 2532, 232, 0

Offset: 0

Ilya Gutkovskiy, Jun 10 2017

			Square array begins:
1,   1,    1,    1,     1,     1,  ...
0,   2,    4,    6,     8,    10,  ...
0,   4,   12,   24,    40,    60,  ...
0,   8,   32,   80,   160,   280,  ...
0,  14,   76,  234,   552,  1110,  ...
0,  24,  168,  624,  1712,  3913,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=0-24 give: A000007, A015128, A001934, A004404 (alternating values), A284286, A004406-A004425 (alternating values).
Rows n=0-2 give: A000012, A005843, A046092.
Main diagonal gives A270919.
Antidiagonal sums give A299108.
Cf. A122141, A286815.

# JacobiTheta4 is defined in A002448.
A288515Column(k, len) = JacobiTheta4(len, -k)
for k in 0:8 A288515Column(k, 8) |> println end # Peter Luschny, Mar 12 2018

Table[Function[k, SeriesCoefficient[Product[((1 + x^i)/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/EllipticTheta[4, 0, x]^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
G.f. of column k: 1/theta_4(x)^k, where theta_4() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A001934.

A303382 Expansion of Product_{n>=1} ((1 + 8x^n)/(1 - 8x^n))^(1/8).

Original entry on oeis.org

1, 2, 4, 50, 98, 1830, 4576, 83950, 236500, 4211766, 12903260, 222377926, 723722602, 12136867530, 41382435824, 678060771778, 2400028798290, 38546050682278, 140724756748476, 2220907298526934, 8323586858891766, 129340015891714962, 495838256186203600

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Expansion of Product_{n>=1} ((1 + 2^b*x^n)/(1 - 2^b*x^n))^(1/(2^b)): A015128 (b=0), A303346 (b=1), A303360 (b=2), this sequence (b=3).

Cf. A303381.

seq(coeff(series(mul(((1+8*x^k)/(1-8*x^k))^(1/8), k = 1..n), x, n+1), x, n), n=0..25); # Muniru A Asiru, Apr 23 2018

nmax = 25; CoefficientList[Series[Product[((1 + 8*x^k)/(1 - 8*x^k))^(1/8), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 23 2018 *)
nmax = 30; CoefficientList[Series[(-7*QPochhammer[-8, x] / (9*QPochhammer[8, x]))^(1/8), {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 23 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+8*x^k)/(1-8*x^k))^(1/8)))

a(n) ~ c * 8^n / n^(7/8), where c = (QPochhammer[-1, 1/8] / QPochhammer[1/8])^(1/8) / Gamma(1/8) = 0.15003359366795844474467456149... - Vaclav Kotesovec, Apr 23 2018

cp's OEIS Frontend

A002132 Generalized sum of divisors function.

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A060046 Generalized sum of divisors function: third diagonal of A060047.

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A261647 Expansion of Product_{k>=0} ((1+x^(2*k+1))/(1-x^(2*k+1)))^3.

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A265844 Expansion of Product_{k>=1} (1 + k^2*x^k)/(1 - k^2*x^k).

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A266821 Expansion of Product_{k>=1} (1 + 3*x^k) / (1 - x^k).

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A278690 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2 in powers of x.

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A284592 Square array read by antidiagonals: T(n,k) is the number of pairs of partitions of n and k respectively, such that the pair of partitions have no part in common.

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A285458 Expansion of Product_{k>=1} ((1 + x^k) / (1 - x^(4*k)))^k.

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A261647 Expansion of Product_{k>=0} ((1+x^(2k+1))/(1-x^(2k+1)))^3.

A265844 Expansion of Product_{k>=1} (1 + k^2x^k)/(1 - k^2x^k).

A303382 Expansion of Product_{n>=1} ((1 + 8x^n)/(1 - 8x^n))^(1/8).