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A215862
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Number of simple labeled graphs on n+2 nodes with exactly n connected components that are trees or cycles.
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9
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0, 4, 19, 55, 125, 245, 434, 714, 1110, 1650, 2365, 3289, 4459, 5915, 7700, 9860, 12444, 15504, 19095, 23275, 28105, 33649, 39974, 47150, 55250, 64350, 74529, 85869, 98455, 112375, 127720, 144584, 163064, 183260, 205275, 229215, 255189, 283309, 313690, 346450
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: (x-4)*x/(x-1)^5.
a(n) = C(n+2,3)*(3*n+13)/4.
a(n) = 5*a(n-1)- 10*a(n-2)+ 10*a(n-3) -5*a(n-4)+a(n-5), n>4. - Harvey P. Dale, Sep 10 2012
a(n) = 1/n! * Sum_{j=0..n} C(n,j)*(-1)^(n-j)*(j)^(n+1)*(j-1)). - Vladimir Kruchinin, Jun 06 2013
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EXAMPLE
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a(1) = 4:
.1-2. .1-2. .1-2. .1 2.
.|/ . .|. . . / . .|/ .
.3... .3... .3... .3...
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MAPLE
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a:= n-> binomial(n+2, 3)*(3*n+13)/4:
seq(a(n), n=0..40);
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MATHEMATICA
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Table[Binomial[n+2, 3] (3n+13)/4, {n, 0, 40}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {0, 4, 19, 55, 125}, 40] (* Harvey P. Dale, Sep 10 2012 *)
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CROSSREFS
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Regarding the sixth formula, see similar sequences listed in A241765.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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