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A050353
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Number of 5-level labeled linear rooted trees with n leaves.
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4
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1, 1, 9, 121, 2169, 48601, 1306809, 40994521, 1469709369, 59277466201, 2656472295609, 130952452264921, 7042235448544569, 410269802967187801, 25740278881968596409, 1730295054262416751321, 124066865052334175027769
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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LINKS
| Index entries for sequences related to rooted trees
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FORMULA
| E.g.f.: (4-3*e^x)/(5-4*e^x).
a(n) is asymptotic to (1/20)*n!/log(5/4)^(n+1). More generally if m>1, the number of m-level labeled linear rooted trees with n leaves is asymptotic to n!/log(m/(m-1))^(n+1)/(m^2-m). - Benoit Cloitre, Jan 30 2003
For m-level trees (m>1), e.g.f. is (m-1-(m-2)*e^x)/(m-(m-1)*e^x) and number of trees is 1/(m*(m-1))*sum(k>=0, (1-1/m)^k*k^n). Here m=5, so a(n)=(1/20)*sum(k>=0, (4/5)^k*k^n) (for n>0). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 30 2003
Let f(x) = (1+x)*(1+2*x). Let D be the operator g(x) -> d/dx(f(x)*g(x)). Then for n>=1, a(n) = D^(n-1)(1) evaluated at x = 3/2. Compare with the result A000670(n) = D^(n-1)(1) at x = 0. See also A194649. - Peter Bala, Sep 05 2011
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MATHEMATICA
| max = 16; f[x_] := (4-3*E^x) / (5-4*E^x); CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]! (* From Jean-François Alcover, Nov 14 2011, after g.f. *)
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PROG
| (PARI) a(n)=n!*if(n<0, 0, polcoeff((4-3*exp(x))/(5-4*exp(x))+O(x^(n+1)), n))
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CROSSREFS
| Cf. A000670, A050351-A050359.
Equals 1/4 * A094417(n) for n>0.
Sequence in context: A046184 A084769 A202835 * A112941 A045976 A053889
Adjacent sequences: A050350 A050351 A050352 * A050354 A050355 A050356
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net), Oct 15 1999.
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