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

Author

N. J. A. Sloane, Mar 19 2001

Keywords

• nonn
• tabf
• easy
• nice

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

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

Cf. A060043, A060044, A060177, A060184.

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

Extensions

More terms from Naohiro Nomoto, Jan 24 2002