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Search: id:A001005
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| A001005 |
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Number of ways of partitioning n points on a circle into subsets only of sizes 2 and 3.
(Formerly M1353 N0520)
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+0
1
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| 1, 0, 1, 1, 2, 5, 8, 21, 42, 96, 222, 495, 1177, 2717, 6435, 15288, 36374, 87516, 210494, 509694, 1237736, 3014882, 7370860, 18059899, 44379535, 109298070, 269766655, 667224480, 1653266565, 4103910930, 10203669285, 25408828065
(list; graph; listen)
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OFFSET
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0,5
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REFERENCES
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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).
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
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LINKS
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INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 396
L. Smiley, a(7) and a(8)
Index entries for sequences related to rooted trees
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FORMULA
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G.f. for a(n+1) satisfies A(x)=x*(1+A(x)^2+A(x)^3). (Bower)
a(n)=sum(((n)!/(k!*j!*(n-k-j+1)!)*[2*k+3*j=n], k=0..floor(n/2), j=0..floor(n/3)). - Len Smiley (smiley(AT)mazzy.math.uaa.alaska.edu), Jun 18 2005
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EXAMPLE
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a(7)=21: 7 rotations of [12][34][567], 7 rotations of [12][45][367], 7 rotations of [12][37][456]
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MAPLE
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a:=proc(n::nonnegint) local k, j; a(n):=0; for k from 0 to floor(n/2) do for j from 0 to floor(n/3) do if (2*k+3*j=n) then a(n):=a(n)+(n)!/(k!*j!*(n-k-j+1)!) fi od od; print(a(n)) end proc; seq(a(i), i=0..30); (Smiley)
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CROSSREFS
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Adjacent sequences: A001002 A001003 A001004 this_sequence A001006 A001007 A001008
Sequence in context: A087077 A117647 A121568 this_sequence A009735 A137095 A092097
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KEYWORD
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nonn,eigen
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms, formula and comment from Christian G. Bower (bowerc(AT)usa.net), Dec 15 1999.
Additional comments from Len Smiley (smiley(AT)mazzy.math.uaa.alaska.edu), Jun 18 2005
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