A002128 MacMahon's generalized sum of divisors function. (Formerly M2784 N1119)

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

6,2

References

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).

Links

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?

Formula

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

Example

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

Prog

(PARI) {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 */

Crossrefs

A diagonal of A060043.

Sequence in context: A318807 A063586 A131477 * A379177 A373170 A064808

Adjacent sequences: A002125 A002126 A002127 * A002129 A002130 A002131

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Sep 21 2007

Status

approved