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Search: id:A002133
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| A002133 |
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Number of partitions of n using only 2 types of parts.
(Formerly M1324 N0507)
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+0
9
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| 0, 0, 1, 2, 5, 6, 11, 13, 17, 22, 27, 29, 37, 44, 44, 55, 59, 68, 71, 81, 82, 102, 97, 112, 109, 136, 126, 149, 141, 168, 157, 188, 176, 212, 182, 231, 207, 254, 230, 266, 241, 300, 259, 319, 283, 344, 295, 373, 311, 386, 352, 417, 353, 452, 368, 460, 418, 492, 413
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Generalized sum of divisors function.
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REFERENCES
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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.
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FORMULA
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G.f.=sum(sum(x^(i+j)/[(1-x^i)(1-x^j)], j=1..i-1), i=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
G.f.: (G(x)^2-H(x))/2 where G(x) = Sum {k>0} x^k/(1-x^k) and H(x) = Sum {k>0} x^(2*k)/(1-x^k)^2. More generally, we obtain g.f. for number of partitions of n with m types of parts if we substitute x(i) with -Sum_{k>0}(x^n/(x^n-1))^i in cycle index Z(S(m); x(1),x(2),..,x(m)) of symmetric group S(m) of degree m. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 18 2007
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EXAMPLE
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a(8)=13 because we have 71, 62, 611, 53, 5111, 422, 41111, 332, 3311, 311111, 22211, 221111, 2111111.
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MAPLE
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g:=sum(sum(x^(i+j)/(1-x^i)/(1-x^j), j=1..i-1), i=1..80): gser:=series(g, x=0, 65): seq(coeff(gser, x^n), n=1..60); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2006
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CROSSREFS
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A diagonal of A060177.
Cf. A002134.
Sequence in context: A140144 A030130 A045845 this_sequence A092306 A090552 A024520
Adjacent sequences: A002130 A002131 A002132 this_sequence A002134 A002135 A002136
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KEYWORD
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nonn,easy
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AUTHOR
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njas
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