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A005380
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Number of partitions of n objects of 2 colors.
(Formerly M1601)
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21
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1, 2, 6, 14, 33, 70, 149, 298, 591, 1132, 2139, 3948, 7199, 12894, 22836, 39894, 68982, 117948, 199852, 335426, 558429, 922112, 1511610, 2460208, 3977963, 6390942, 10206862, 16207444, 25596941, 40214896, 62868772, 97814358
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. A. MacMahon, Memoir on symmetric functions of the roots of systems of equations, Phil. Trans. Royal Soc. London, 181 (1890), 481-536; Coll. Papers II, 32-87.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Theory and application of plane partitions II, Studies in Appl. Math., 50 (1971), 259-279.
R. P. Stanley, Conjugate trace..., J. Combin. Theory, vol. A14 53-65 1973, esp. p. 64.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.99.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
N. J. A. Sloane, Transforms
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FORMULA
| G.f.: Product_{k=1..infinity} 1/(1-x^k)^(k+1). EULER transform of b(n) = n+1.
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MAPLE
| mul( (1-x^i)^(-i-1), i=1..80); series(%, x, 80); seriestolist(%);
with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr (n-> n+1): seq (a(n), n=0..31); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 08 2008]
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MATHEMATICA
| max = 31; f[x_] = Product[ 1/(1-x^k)^(k+1), {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* From Jean-François Alcover, Nov 08 2011, after g.f. *)
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PROG
| (PARI) a(n)=polcoeff(prod(i=1, n, (1-x^i+x*O(x^n))^-(i+1)), n)
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CROSSREFS
| Row sums of A054225. Cf. A000219.
Sequence in context: A110524 A083404 A089351 * A124612 A184697 A124613
Adjacent sequences: A005377 A005378 A005379 * A005381 A005382 A005383
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Christian G. Bower (bowerc(AT)usa.net), Sep 07 2002
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