A385001 -id:A385001 - cp's OEIS frontend

A002127 MacMahon's generalized sum of divisors function.

1, 3, 9, 15, 30, 45, 67, 99, 135, 175, 231, 306, 354, 465, 540, 681, 765, 945, 1040, 1305, 1386, 1695, 1779, 2205, 2290, 2754, 2835, 3438, 3480, 4185, 4272, 5076, 5004, 6100, 5985, 7155, 7154, 8325, 8190, 9840, 9471, 11241, 11055, 12870, 12420, 14911

Offset: 3

N. J. A. Sloane

Number of partitions of n with two designated summands. For example: a(5) = 9 because there are 9 partitions of 5 with two designated summands: [4'+ 1'], [3'+ 2'], [3'+ 1'+ 1], [3'+ 1 + 1'], [2'+ 2 + 1'], [2 + 2'+ 1'], [2'+ 1'+ 1 + 1], [2'+ 1 + 1'+ 1], [2'+ 1 + 1 + 1']. - Omar E. Pol, Jul 23 2025

			x^3 + 3*x^4 + 9*x^5 + 15*x^6 + 30*x^7 + 45*x^8 + 67*x^9 + 99*x^10 + ...

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).

John Cerkan, Table of n, a(n) for n = 3..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.
William Craig, Jan-Willem van Ittersum and Ken Ono, Integer partitions detect the primes, PNAS, Vol. 121, No. 39 (2024), e2409417121.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
S. Rose, What literature is known about MacMahon's generalized sum-of-divisors function?

A diagonal of A060043.
Cf. A002128.
Column 2 of A385001.

A002127[n_] := (DivisorSigma[3, n] - (2*n - 1)*DivisorSigma[1, n])/8;
Array[A002127, 50, 3] (* Paolo Xausa, Jul 04 2025, after Michael Somos's PARI *)

{a(n) = if( n<1, 0, ( sigma( n, 3) - (2*n - 1) * sigma(n) ) / 8)} /* Michael Somos, Jan 10 2012 */

G.f.: (Sum_{k>=0} (-1)^k * (2*k + 1) * binomial( k+2, 4) * x^( k*(k+1) / 2 )) / (5  * Sum_{k>=0} (-1)^k * (2*k + 1) * x^( k*(k+1) / 2 )). - Michael Somos, Jan 10 2012
a(n) = (n^2 - 3*n + 2) * A000203(n) / 8 iff n is an odd prime (see Craig link et al.).
Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / (4!*5!). - Vaclav Kotesovec, Aug 01 2025

More terms from Vladeta Jovovic, Nov 11 2001

A002128 MacMahon's generalized sum of divisors function.

Original entry on oeis.org

1, 3, 9, 22, 42, 81, 140, 231, 351, 551, 783, 1134, 1546, 2142, 2835, 3758, 4818, 6237, 7826, 9885, 12159, 14974, 18261, 22113, 26511, 31668, 37611, 44149, 52074, 60660, 70569, 81396, 94311, 107317, 123879, 140049, 160154, 179949, 204867, 228137

Offset: 6

N. J. A. Sloane

Number of partitions of n with three designated summands. For example: a(8) = 9 because there are 9 partitions of 8 with three designated summands: [5'+ 2'+ 1'], [4'+ 3'+ 1'], [4'+ 2'+ 1'+ 1], [4'+ 2'+ 1 + 1'], [3'+ 2'+ 2 + 1'], [3'+ 2 + 2'+ 1'], [3'+ 2'+ 1'+ 1 + 1], [3'+ 2'+ 1 + 1'+ 1], [3'+ 2'+ 1 + 1 + 1']. - Omar E. Pol, Jul 25 2025

			x^6 + 3*x^7 + 9*x^8 + 22*x^9 + 42*x^10 + 81*x^11 + 140*x^12 + 231*x^13 + ...

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).

John Cerkan, Table of n, a(n) for n = 6..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 (1921), 75-113; Coll. Papers II, pp. 303-341.
S. Rose, What literature is known about MacMahon's generalized sum-of-divisors function?

A diagonal of A060043.
Cf. A002127.
Column 3 of A385001.

{a(n) = if( n<1, 0, (3*sigma(n,5) + (-30*n + 50)*sigma(n,3) + (40*n^2 - 100*n + 37)*sigma(n)) / 1920)} /* Michael Somos, Jan 10 2012 */

G.f.: (t(1)^3-3*t(1)*t(2)+2*t(3))/6 where t(i) = Sum(x^(n*i)/(1-x^n)^(2*i),n=1..inf), i=1..3. - Vladeta Jovovic, Sep 21 2007
G.f.: (Sum_{k>=0} (-1)^k * (2*k + 1) * binomial( k+3, 6) * x^( k*(k+1) / 2 )) / (-7  * Sum_{k>=0} (-1)^k * (2*k + 1) * x^( k*(k+1) / 2 )). - Michael Somos, Jan 10 2012
Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / (6!*7!). - Vaclav Kotesovec, Aug 01 2025

More terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Sep 21 2007

A365664 Expansion of Sum_{0

Original entry on oeis.org

1, 3, 9, 22, 51, 97, 188, 330, 568, 918, 1452, 2233, 3344, 4884, 7004, 9856, 13653, 18699, 25080, 33462, 43918, 57304, 73668, 94482, 119262, 150285, 187231, 232560, 285660, 350746, 425627, 516477, 620731, 745503, 887796, 1056669, 1247521, 1472460, 1726054, 2021327

Offset: 10

Seiichi Manyama, Sep 15 2023

nonn

Number of partitions of n with four designated summands. For example: a(11) = 3 because there are three partitions of 11 with four designated summands: [5'+ 3'+ 2'+ 1'], [4'+ 3'+ 2'+ 1'+ 1], [4'+ 3'+ 2'+ 1 + 1']. - Omar E. Pol, Jul 26 2025

Seiichi Manyama, Table of n, a(n) for n = 10..10000
George E. Andrews and Simon C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms, Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 2013, No. 676 (2013), pp. 97-103; arXiv preprint, arXiv:1010.5769 [math.NT], 2010.

A diagonal of A060043.
Cf. A000203, A002127, A002128, A365665.
Cf. A010816, A365630, A365666.
Cf. A001158, A001160, A013955.
Column k=4 of A385001.

a[n_] := Module[{d = DivisorSigma[{1, 3, 5, 7}, n]}, (5*d[[4]] - (126*n-441)*d[[3]] + (756*n^2-4410*n+4935)*d[[2]] - (840*n^3-5880*n^2+9870*n-3229)*d[[1]])/967680]; Array[a, 40, 10] (* Amiram Eldar, Jan 07 2025 *)

a(n) = (5*sigma(n, 7)-(126*n-441)*sigma(n, 5)+(756*n^2-4410*n+4935)*sigma(n, 3)-(840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680; \\ Seiichi Manyama, Jul 24 2024

G.f.: (1/9) * ( Sum_{k>=4} (-1)^k * (2*k+1) * binomial(k+4,8) * q^(k*(k+1)/2) ) / ( Sum_{k>=0} (-1)^k * (2*k+1) * q^(k*(k+1)/2) ).
a(n) = (5*sigma_7(n) - (126*n-441)*sigma_5(n) + (756*n^2-4410*n+4935)*sigma_3(n) - (840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680. - Seiichi Manyama, Jul 24 2024
Sum_{k=1..n} a(k) ~ Pi^8 * n^8 / (8!*9!). - Vaclav Kotesovec, Aug 01 2025

A365665 Expansion of Sum_{0

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 208, 390, 693, 1193, 1977, 3195, 4995, 7722, 11583, 17164, 24882, 35685, 50205, 70083, 96300, 131101, 176358, 235377, 310651, 407352, 529074, 682750, 874038, 1112085, 1405521, 1766259, 2206413, 2741431, 3389052, 4168089, 5103450, 6218469

Offset: 15

Seiichi Manyama, Sep 15 2023

nonn

Number of partitions of n with five designated summands (when part i has multiplicity j > 0 exactly one part i is "designated"). For example: a(16) = 3 because there are three partitions of 16 with five designated summands: [6'+ 4'+ 3'+ 2'+ 1'], [5'+ 4'+ 3'+ 2'+ 1'+ 1], [5'+ 4'+ 3'+ 2'+ 1 + 1']. - Omar E. Pol, Jul 29 2025

Vaclav Kotesovec, Table of n, a(n) for n = 15..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.

A diagonal of A060043.
Cf. A000203, A002127, A002128, A365664.
Cf. A010816, A365631, A365667.
Column k=5 of A385001.
Cf. A384926.

nmax = 60; Drop[CoefficientList[Series[-1/11 * Sum[(-1)^k*(2*k + 1)*Binomial[k + 5, 10]*x^(k*(k + 1)/2), {k, 5, nmax}]/Sum[(-1)^k*(2*k + 1)*x^(k*(k + 1)/2), {k, 0, nmax}], {x, 0, nmax}], x], 15] (* Vaclav Kotesovec, Jul 29 2025 *)
(* or *)
Table[(10679/17203200 - 1571*n/774144 + 133*n^2/92160 - n^3/3072 + n^4/46080) * DivisorSigma[1, n] + (1571/1548288 - 133*n/122880 + 3*n^2/10240 - n^3/46080) * DivisorSigma[3, n] + (133/1228800 - n/20480 + n^2/215040) * DivisorSigma[5, n] + (1/516096 - n/3096576) * DivisorSigma[7, n] + DivisorSigma[9, n]/154828800, {n, 15, 60}] (* Vaclav Kotesovec, Jul 29 2025 *)

G.f.: -(1/11) * ( Sum_{k>=5} (-1)^k * (2*k+1) * binomial(k+5,10) * q^(k*(k+1)/2) ) / ( Sum_{k>=0} (-1)^k * (2*k+1) * q^(k*(k+1)/2) ).
From Vaclav Kotesovec, Jul 29 2025: (Start)
a(n) = (10679/17203200 - 1571*n/774144 + 133*n^2/92160 - n^3/3072 + n^4/46080)*sigma(n) + (1571/1548288 - 133*n/122880 + 3*n^2/10240 - n^3/46080)*sigma_3(n) + (133/1228800 - n/20480 + n^2/215040)*sigma_5(n) + (1/516096 - n/3096576)*sigma_7(n) + sigma_9(n)/154828800.
Sum_{k=1..n} a(k) ~ Pi^10 * n^10 / 144850083840000.
(End)

A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

Original entry on oeis.org

1, -1, -3, -3, -4, 3, 2, 19, 21, 32, 40, 45, 16, 8, -18, -125, -164, -291, -358, -530, -588, -724, -592, -675, -358, -207, 570, 1201, 2208, 3333, 4944, 6490, 8277, 10492, 11800, 13260, 14328, 14722, 12942, 12075, 5640, 603, -10444, -21120, -39360, -55876, -83488

Offset: 0

Ilya Gutkovskiy, Nov 06 2019

sign

Convolution inverse of A329156.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A003056, A032198, A077285, A088305, A104575, A319668, A329156, A385001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1, b(n, i-1), 0)-
      add(b(n-i*j, min(n-i*j, i-1))*j, j=`if`(i=1, n, 1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..46);  # Alois P. Heinz, Jul 18 2025

nmax = 46; CoefficientList[Series[Product[(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 46; CoefficientList[Series[Product[(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k / (1 - x^k)^2).
G.f.: exp(-Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).
G.f.: Product_{k>=1} (1 - x^k)^A032198(k).
G.f.: A(x) = Product_{k>=1} 1 / B(x^k), where B(x) = g.f. of A088305.
a(n) = Sum_{k=0..A003056(n)} (-1)^k * A385001(n,k). - Alois P. Heinz, Jul 18 2025

A384926 Number of partitions of n with six designated summands.

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 221, 414, 765, 1344, 2310, 3834, 6248, 9894, 15408, 23550, 35394, 52353, 76402, 109959, 156366, 219850, 305796, 421281, 574568, 777234, 1042083, 1387037, 1831362, 2402595, 3128995, 4051797, 5211639, 6668490, 8482089, 10737063, 13516615

Offset: 21

Omar E. Pol, Jul 23 2025

nonn

			21 has only one partition with six designated summands: [6'+ 5'+ 4'+ 3'+ 2'+ 1'], so a(21) = 1.
22 has three partitions with six designated summands: [7'+ 5'+ 4'+ 3'+ 2'+ 1'], [6'+ 5'+ 4'+ 3'+ 2'+ 1'+ 1], [6'+ 5'+ 4'+ 3'+ 2'+ 1 + 1'], so a(22) = 3.

Alois P. Heinz, Table of n, a(n) for n = 21..4000

Column k=6 of A385001.
Columns of A385001 converge to A000716.
Other columns k of A385001 are A000007 (k=0), A000203 (k=1), A002127 (k=2), A002128 (k=3), A365664 (k=4), A365665 (k=5).

b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*j*x, j=1..n/i))), x, 7)
    end:
a:= n-> coeff(b(n$2), x, 6):
seq(a(n), n=21..57);  # Alois P. Heinz, Jul 23 2025

nmax=60; Drop[CoefficientList[Series[1/13 * Sum[(-1)^k*(2*k + 1)*Binomial[k + 6, 12]*x^(k*(k + 1)/2), {k, 6, nmax}]/Sum[(-1)^k*(2*k + 1)*x^(k*(k + 1)/2), {k, 0, nmax}], {x, 0, nmax}], x] , 21] (* Vaclav Kotesovec, Jul 29 2025 *)

A000716(n) >= a(21+n) with equality only for n <= 6.

Sum_{k=1..n} a(k) ~ Pi^12 * n^12 / (12! * 13!). - Vaclav Kotesovec, Aug 01 2025

More terms from Alois P. Heinz, Jul 23 2025

A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 8, 1, 1, 15, 4, 1, 21, 13, 1, 33, 28, 1, 1, 41, 58, 4, 1, 56, 103, 13, 1, 69, 170, 35, 1, 87, 269, 77, 1, 1, 99, 404, 158, 4, 1, 127, 579, 298, 13, 1, 141, 810, 529, 35, 1, 165, 1116, 880, 86, 1, 189, 1470, 1431, 183, 1, 1, 220, 1935, 2214, 371, 4, 1, 238, 2475, 3348, 701, 13

Offset: 0

Omar E. Pol, Jul 22 2025

When part i has multiplicity j > 0 exactly one part i is "designated".
The length of the row n is A002024(n+1) = 1 + A003056(n), hence the first positive element in column k is in the row A000217(k).
Column k gives the partial sums of the column k of A385001.
Columns converge to A210843 which is also the partial sums of A000716.

			Triangle begins:
---------------------------------------------
   n\k:   0    1     2      3     4    5   6
---------------------------------------------
   0 |    1;
   1 |    1,   1;
   2 |    1,   4;
   3 |    1,   8,    1;
   4 |    1,  15,    4;
   5 |    1,  21,   13;
   6 |    1,  33,   28,     1;
   7 |    1,  41,   58,     4;
   8 |    1,  56,  103,    13;
   9 |    1,  69,  170,    35;
  10 |    1,  87,  269,    77,    1;
  11 |    1,  99,  404,   158,    4;
  12 |    1, 127,  579,   298,   13;
  13 |    1, 141,  810,   529,   35;
  14 |    1, 165, 1116,   880,   86;
  15 |    1, 189, 1470,  1431,  183,   1;
  16 |    1, 220, 1935,  2214,  371,   4;
  17 |    1, 238, 2475,  3348,  701,  13;
  18 |    1, 277, 3156,  4894, 1269,  35;
  19 |    1, 297, 3921,  7036, 2187,  86;
  20 |    1, 339, 4866,  9871, 3639, 194;
  21 |    1, 371, 5906, 13629, 5872, 402,  1;
  ...

Alois P. Heinz, Rows n = 0..1000, flattened

Columns: A000012 (k=0), A024916 (k=1).

Cf. A000217, A000716, A002024, A003056, A077285, A195825, A210843, A211970, A385001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
    end:
g:= proc(n) option remember; `if`(n<0, 0, g(n-1)+b(n$2)) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 22 2025

A384998 Total number of partitions of all numbers <= n with designated summands, n >= 0.

Original entry on oeis.org

1, 2, 5, 10, 20, 35, 63, 104, 173, 275, 435, 666, 1018, 1516, 2248, 3275, 4745, 6776, 9632, 13528, 18910, 26182, 36078, 49311, 67111, 90690, 122052, 163271, 217559, 288350, 380806, 500504, 655601, 855113, 1111777, 1439911, 1859347, 2392509, 3069921, 3926494

Offset: 0

Omar E. Pol, Aug 06 2025

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Partial sums of A077285.
Row sums of A384999.
Cf. A385001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1)*j, j=1..n/i)))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n$2)) end:
seq(a(n), n=0..41);  # Alois P. Heinz, Aug 06 2025

nmax = 50; CoefficientList[Series[1/(1-x) * Product[(1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 08 2025 *)

From Vaclav Kotesovec, Aug 08 2025: (Start)
a(n) ~ 5^(1/4) * exp(sqrt(10*n)*Pi/3) / (2^(9/4) * sqrt(3) * Pi * n^(3/4)).
G.f.: 1/(1-x) * Product_{k>=1} (1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))). (End)

cp's OEIS Frontend

A002127 MacMahon's generalized sum of divisors function.

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A002128 MacMahon's generalized sum of divisors function.

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PARI

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A365664 Expansion of Sum_{0

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A365665 Expansion of Sum_{0

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A329157 Expansion of Product_{k>=1} (1 - Sum_{j>=1} j * x^(k*j)).

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Maple

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A384926 Number of partitions of n with six designated summands.

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Maple

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A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).

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