|
Search: id:A002133
|
|
|
| A002133 |
|
Number of partitions of n using only 2 types of parts.
(Formerly M1324 N0507)
|
|
+0
9
|
|
| 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)
|
|
|
OFFSET
|
1,4
|
|
|
COMMENT
|
Generalized sum of divisors function.
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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.
|
|
FORMULA
|
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.rs), Sep 18 2007
|
|
EXAMPLE
|
a(8)=13 because we have 71, 62, 611, 53, 5111, 422, 41111, 332, 3311, 311111, 22211, 221111, 2111111.
|
|
MAPLE
|
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
|
|
CROSSREFS
|
A diagonal of A060177.
Cf. A002134.
Sequence in context: A030130 A164874 A045845 this_sequence A092306 A090552 A024520
Adjacent sequences: A002130 A002131 A002132 this_sequence A002134 A002135 A002136
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
Search completed in 0.005 seconds
|
