A060047 Triangle of generalized sum of divisors function, read by rows.

1, 2, 4, 1, 4, 2, 6, 4, 8, 8, 8, 14, 8, 1, 18, 13, 2, 28, 12, 4, 40, 12, 8, 52, 16, 14, 70, 14, 24, 88, 16, 40, 104, 24, 1, 56, 140, 16, 2, 84, 168, 18, 4, 122, 196, 26, 8, 168, 240, 20, 14, 232, 278, 24, 24, 312, 320, 32, 40, 408, 380, 24, 64, 528, 440, 24, 100, 672, 504

Offset

1,2

Comments

Lengths of rows are 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 ... (A000196).

Links

Table of n, a(n) for n=1..69.

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.

Formula

T(n, k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*(2*m_1-1) + s_2*(2*m_2-1) + ... + s_k*(2*m_k-1) = 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^(2*m_1+2*m_2+...+2*m_k-k)/((1-q^{2*m_1-1})*(1-q^{2*m_2-1})*...*(1-q^{2*m_k-1}))^2 = Sum_n T(n, k)*q^n.

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

Example

Triangle turned on its side begins:
  1  2  4  4  6  8  8  8 13 12 12 ...
           1  2  4  8 14 18 28 40 ...
                          1  2  4 ...
For example, T(6,1) = 8, T(6,2) = 4.

Crossrefs

Diagonals give A002131, A002132, A060046, A365666, A365667.

Cf. A060043, A060044, A060177, A060184.

Sequence in context: A011167 A014176 A317690 * A135185 A289460 A201774

Adjacent sequences: A060044 A060045 A060046 * A060048 A060049 A060050

Keyword

nonn,tabf,easy,nice

Author

N. J. A. Sloane, Mar 19 2001

Extensions

More terms from Naohiro Nomoto, Jan 24 2002

Status

approved