A060044 Triangle of generalized sum of divisors function, read by rows.

1, -1, 1, 4, -1, -5, 1, 6, 1, 3, -4, -1, -2, 8, 1, 1, -13, -2, -5, 13, 1, 10, 23, -6, -1, -11, -25, 12, 1, 12, 27, -20, -2, -21, -49, 14, 3, 31, 74, -8, 1, 5, -13, -62, 24, -1, -4, 23, 85, -29, 1, 2, -42, -132, 18, -2, -8, 42, 165, -13, 3, 14, -42, -195, 20, -4, -20, 43, 229, -30

Offset: 1

Author

N. J. A. Sloane, Mar 19 2001

Keywords

• sign
• tabf
• easy
• nice

Comments

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

Examples

			Triangle turned on its side begins:
  1  -1   4  -5   6  -4   8 -13  13 ...
          1  -1   1   3  -2   1  -5 ...
                      1  -1   1  -2 ...
For example, T(8,3) = 1.
		

Links

• 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 A002129, A002130, A060045. Cf. A060043, A060177.

Cf. A003056.

Formula

T(n, k) = sum of (-1)^(k+s_1+s_2+...+s_k) * 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.

Extensions

More terms from Naohiro Nomoto, Jan 24 2002