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A002128
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MacMahon's generalized sum of divisors function.
(Formerly M2784 N1119)
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 6,2
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REFERENCES
| P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
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).
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LINKS
| G. E. Andrews and S. C. F. Rose, MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms
S. Rose, What literature is known about MacMahon's generalized sum-of-divisors function?
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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 (vladeta(AT)eunet.rs), 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
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EXAMPLE
| x^6 + 3*x^7 + 9*x^8 + 22*x^9 + 42*x^10 + 81*x^11 + 140*x^12 + 231*x^13 + ...
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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 */
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CROSSREFS
| A diagonal of A060043.
Sequence in context: A098980 A063586 A131477 * A064808 A192389 A187053
Adjacent sequences: A002125 A002126 A002127 * A002129 A002130 A002131
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 24 2002
More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 21 2007
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