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A005175 Number of trees of subsets of an n-set.
(Formerly M3173)
1
0, 0, 3, 131, 1830, 16990, 127953, 851361, 5231460, 30459980, 170761503, 931484191, 4979773890, 26223530970, 136522672653, 704553794621, 3611494269120, 18415268221960, 93516225653403, 473366777478651, 2390054857197150, 12043393363764950, 60590148885015753 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=1..23.

F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.

F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for sequences related to trees

FORMULA

3*(3^n-2*2^n+1)/2 + 113*(4^n-3*3^n+3*2^n-1)/6 + 625*(5^n-4*4^n+6*3^n-4*2^n+1)/24. - formula fitted by John W. Layman.

a(n) = (125/24) * 5^n - (64/3) * 4^n + (135/4)*3^n - (76/3) * 2^n + 209/24 proven in McMorris and Zaslavsky, matches Layman's formula with an offset of 1. - Sean A. Irvine, Apr 12 2016

E.g.f.: (1/24)*exp(x)*(-1 + exp(x))^2*(209 - 798*exp(x) + 625*exp(2*x)). - Ilya Gutkovskiy, Apr 12 2016

MAPLE

A005175:=-z**2*(3+86*z+120*z**2)/(z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1); # Conjectured by Simon Plouffe in his 1992 dissertation.

MATHEMATICA

Table[(125/24) 5^n - (64/3) 4^n + (135/4) 3^n - (76/3) 2^n + 209/24, {n, 20}] (* Michael De Vlieger, Apr 12 2016 *)

CROSSREFS

Sequence in context: A202030 A249379 A139943 * A082439 A082622 A075597

Adjacent sequences:  A005172 A005173 A005174 * A005176 A005177 A005178

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 21 09:14 EST 2019. Contains 329362 sequences. (Running on oeis4.)