A002128 -id:A002128

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A060043

Triangle T(n,k), n >= 1, k >= 1, of generalized sum of divisors function, read by rows.

+10
9

1, 3, 1, 4, 3, 7, 9, 6, 1, 15, 12, 3, 30, 8, 9, 45, 15, 22, 67, 13, 1, 42, 99, 18, 3, 81, 135, 12, 9, 140, 175, 28, 22, 231, 231, 14, 51, 351, 306, 24, 1, 97, 551, 354, 24, 3, 188, 783, 465, 31, 9, 330, 1134, 540, 18, 22, 568, 1546, 681, 39, 51, 918, 2142, 765, 20

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Lengths of rows are 1 1 2 2 2 3 3 3 3 ... (A003056).

LINKS

Table of n, a(n) for n=1..65.

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.

FORMULA

T(n, 1) = sum of divisors of n (A000203), T(n, k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*m_1 + s_2*m_2 + ... + s_k*m_k = n and the sum is over all such k-partitions of n.

G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1-q^m_1)*(1-q^m_2)*...*(1-q^m_k))^2 = Sum_n T(n, k)*q^n.

G.f. for k-th diagonal: (-1)^k * (1/(2*k+1)) * ( Sum_{j>=k} (-1)^j * (2*j+1) * binomial(j+k,2*k) * q^(j*(j+1)/2) ) / ( Sum_{j>=0} (-1)^j * (2*j+1) * q^(j*(j+1)/2) ). - Seiichi Manyama, Sep 15 2023

EXAMPLE

Triangle turned on its side begins:

1 3 4 7 6 12 8 15 13 18 ...

1 3 9 15 30 45 67 99 ...

1 3 9 22 42 ...

1 ...

For example, T(6,2) = 15.

MATHEMATICA

Clear[diag, m]; nmax = 19; kmax = Floor[(Sqrt[8*nmax+1]-1)/2]; m[0] = 0; diag[k_] := diag[k] = Sum[q^(Sum[m[i], {i, 1, k}])/(Times @@ (1 - q^Array[m, k]))^2, Sequence @@ Table[{m[j], m[j-1]+1, nmax}, {j, 1, k}] // Evaluate] + O[q]^(nmax+1) // CoefficientList[#, q]&; Table[ Select[ Table[diag[k][[j+1]], {k, 1, kmax}], IntegerQ[#] && # > 0&] // Reverse, {j, 1, nmax}] // Flatten (* Jean-François Alcover, Jul 18 2017 *)

CROSSREFS

Diagonals give A000203, A002127, A002128, A365664, A365665.

Cf. A010816, A060044, A060047.

KEYWORD

nonn,tabf,easy,nice

AUTHOR

N. J. A. Sloane, Mar 19 2001

EXTENSIONS

More terms from Naohiro Nomoto, Jan 24 2002

STATUS

approved

A365664

Expansion of Sum_{0<i<j<k<l} q^(i+j+k+l)/( (1-q^i)*(1-q^j)*(1-q^k)*(1-q^l) )^2.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

10,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 10..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.

FORMULA

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

PROG

(PARI) 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

CROSSREFS

A diagonal of A060043.

Cf. A000203, A002127, A002128, A365665.

Cf. A010816, A365630, A365666.

Cf. A001158, A001160, A013955.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Sep 15 2023

STATUS

approved

A374930

Expansion of Sum_{1<=i<=j<=k} q^(i+j+k)/( (1-q^i)*(1-q^j)*(1-q^k) )^2.

+10
3

1, 7, 27, 77, 181, 378, 707, 1254, 2052, 3290, 4928, 7371, 10381, 14756, 19818, 27158, 35139, 46683, 58806, 76146, 93555, 119092, 143222, 178983, 212408, 261261, 305046, 371931, 428156, 515592, 589385, 701442, 792720, 939918, 1050567, 1233387, 1374835, 1600143, 1766583, 2052898, 2247784

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,2

LINKS

Table of n, a(n) for n=3..43.

Tewodros Amdeberhan, George E. Andrews and Roberto Tauraso, Extensions of MacMahon's sums of divisors, arXiv:2309.03191v1 [math.CO], Sep 06 2023.

FORMULA

a(n) = (31*sigma_5(n) - 70*(n+1)*sigma_3(n) + (40*n^2+60*n+9)*sigma(n))/1920.

MATHEMATICA

A374930[n_] := (31*DivisorSigma[5, n] - 70*(n + 1)*DivisorSigma[3, n] + (40*n^2 + 60*n + 9)*DivisorSigma[1, n])/1920;

Array[A374930, 50, 3] (* Paolo Xausa, Jul 24 2024 *)

PROG

(PARI) a(n) = (31*sigma(n, 5)-70*(n+1)*sigma(n, 3)+(40*n^2+60*n+9)*sigma(n))/1920;

(Python)

from math import prod

from sympy import factorint

def A374930(n):

f = factorint(n).items()

return (31*prod((p**(5*(e+1))-1)//(p**5-1) for p, e in f)-70*(n+1)*prod((p**(3*(e+1))-1)//(p**3-1) for p, e in f) + (20*n*((n<<1)+3)+9)*prod((p**(e+1)-1)//(p-1) for p, e in f))//1920 # Chai Wah Wu, Jul 24 2024

CROSSREFS

Cf. A374929, A374931.

Cf. A000203, A001158, A001160, A002128.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jul 24 2024

STATUS

approved

A365665

Expansion of Sum_{0<i<j<k<l<m} q^(i+j+k+l+m)/( (1-q^i)*(1-q^j)*(1-q^k)*(1-q^l)*(1-q^m) )^2.

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

15,2

LINKS

Table of n, a(n) for n=15..52.

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.

FORMULA

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) ).

CROSSREFS

A diagonal of A060043.

Cf. A000203, A002127, A002128, A365664.

Cf. A010816, A365631, A365667.