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A060281
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Triangle T(n,k) read by rows giving number of labeled mappings (or functional digraphs) from n points to themselves (endofunctions) with exactly k cycles, k=1..n.
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24
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1, 3, 1, 17, 9, 1, 142, 95, 18, 1, 1569, 1220, 305, 30, 1, 21576, 18694, 5595, 745, 45, 1, 355081, 334369, 113974, 18515, 1540, 63, 1, 6805296, 6852460, 2581964, 484729, 49840, 2842, 84, 1, 148869153, 158479488, 64727522, 13591116, 1632099, 116172, 4830, 108, 1
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OFFSET
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1,2
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COMMENTS
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Also called sagittal graphs.
T(n,k)=1 iff n=k (counts the identity mapping of [n]). - Len Smiley, Apr 03 2006
Also the coefficients of the tree polynomials t_{n}(y) defined by (1-T(z))^(-y) = Sum_{n>=0} t_{n}(y) (z^n/n!) where T(z) is Cayley's tree function T(z) = Sum_{n>=1} n^(n-1) (z^n/n!) giving the number of labeled trees A000169. - Peter Luschny, Mar 03 2009
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REFERENCES
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I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
W. Szpankowski. Average case analysis of algorithms on sequences. John Wiley & Sons, 2001. - Peter Luschny, Mar 03 2009
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LINKS
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FORMULA
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E.g.f.: 1/(1 + LambertW(-x))^y.
T(n,k) = Sum_{j=0..n-1} C(n-1,j)*n^(n-1-j)*(-1)^(k+j+1)*A008275(j+1,k) = Sum_{j=0..n-1} binomial(n-1,j)*n^(n-1-j)*s(j+1,k). [Riordan] (Note: s(m,p) denotes signless Stirling cycle number (first kind), A008275 is the signed triangle.) - Len Smiley, Apr 03 2006
Sum_{k=1..n} k * T(n,k) = A190314(n).
Sum_{k=1..n} (-1)^(k+1) * T(n,k) = A000169(n) for n>=1. (End)
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EXAMPLE
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Triangle T(n,k) begins:
: 1;
: 3, 1;
: 17, 9, 1;
: 142, 95, 18, 1;
: 1569, 1220, 305, 30, 1;
: 21576, 18694, 5595, 745, 45, 1;
: 355081, 334369, 113974, 18515, 1540, 63, 1;
: 6805296, 6852460, 2581964, 484729, 49840, 2842, 84, 1;
: ...
T(3,2)=9: (1,2,3)--> [(2,1,3),(3,2,1),(1,3,2),(1,1,3),(1,2,1), (1,2,2),(2,2,3),(3,2,3),(1,3,3)].
Tree polynomials (with offset 0):
t_0(y) = 1;
t_1(y) = y;
t_2(y) = 3y + y^2;
t_3(y) = 17y + 9y^2 + y^3; (End)
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MAPLE
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with(combinat):T:=array(1..8, 1..8):for m from 1 to 8 do for p from 1 to m do T[m, p]:=sum(binomial(m-1, k)*m^(m-1-k)*(-1)^(p+k+1)*stirling1(k+1, p), k=0..m-1); print(T[m, p]) od od; # Len Smiley, Apr 03 2006
T := z -> sum(n^(n-1)*z^n/n!, n=1..16):
p := convert(simplify(series((1-T(z))^(-y), z, 12)), 'polynom'):
seq(print(coeff(p, z, i)*i!), i=0..8); (End)
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MATHEMATICA
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t=Sum[n^(n-1) x^n/n!, {n, 1, 10}];
Transpose[Table[Rest[Range[0, 10]! CoefficientList[Series[Log[1/(1 - t)]^n/n!, {x, 0, 10}], x]], {n, 1, 10}]]//Grid (* Geoffrey Critzer, Mar 13 2011*)
Table[k! SeriesCoefficient[1/(1 + ProductLog[-t])^x, {t, 0, k}, {x, 0, j}], {k, 10}, {j, k}] (* Jan Mangaldan, Mar 02 2013 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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