|
| |
|
|
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
|
| |
|
|
