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A066324
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Number of endofunctions on n labeled points constructed from k rooted trees.
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3
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1, 2, 2, 9, 12, 6, 64, 96, 72, 24, 625, 1000, 900, 480, 120, 7776, 12960, 12960, 8640, 3600, 720, 117649, 201684, 216090, 164640, 88200, 30240, 5040, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320, 43046721
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| T(n,k) = number of endofunctions with k recurrent elements. - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 06 2006
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REFERENCES
| F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 87, see (2.3.28).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.32.
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FORMULA
| T(n, k)=k*n^(n-k)*(n-1)!/(n-k)!
E.g.f. (relative to x): A(x, y)=1/(1-y*B(x)) - 1 = y*x +(2*y+2*y^2)*x^2/2! + (9*y+12*y^2+6*y^3)*x^3/3! + ..., where B(x) is e.g.f. A000169.
From Peter Bala, Sep 30 2011: (Start)
Let F(x,t) = x/(1+t*x)*exp(-x/(1+t*x)) = x*(1 - (1+t)*x + (1+4*t+2*t^2)*x^2/2! - ...). F is essentially the e.g.f. for A144084 (see also A021010). Then the e.g.f. for the present table is t*F(x,t)^(-1), where the compositional inverse is taken with respect to x.
Removing a factor of n from the n-th row entries results in A122525 in row reversed form.
(End)
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EXAMPLE
| 1; 2,2; 9,12,6; 64,96,72,24; 625,1000,900,480,120; ...
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MATHEMATICA
| f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Flatten[Map[f, Drop[Range[0, 10]! CoefficientList[Series[1/(1 - y*t), {x, 0, 10}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 05 2011 *)
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PROG
| (PARI) T(n, k)=k*n^(n-k)*(n-1)!/(n-k)! \\ Charles R Greathouse IV, Dec 05 2011
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CROSSREFS
| Column 1: A000169. Main diagonal: A000142. T(n, n-1): A062119. Row sums give A000312. A021010, A122525, A144084.
Sequence in context: A143022 A154100 A002880 * A143146 A185755 A039796
Adjacent sequences: A066321 A066322 A066323 * A066325 A066326 A066327
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KEYWORD
| nonn,tabl
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Dec 14 2001
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