OFFSET
1,3
REFERENCES
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
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; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (6, -11, 6).
FORMULA
G.f.: x*(1 + 6*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)). [corrected by Ray Chandler, Jun 26 2023]
First differences give A003063, 3^(n-1) - 2^n.
From Andrew Howroyd, Mar 28 2025: (Start)
a(n) = (3^(n+1) - 2^(n+3) + 7)/2.
E.g.f.: (3*exp(x)/2 - 1)*(exp(x) - 1)^2. (End)
EXAMPLE
From Andrew Howroyd, Mar 28 2025: (Start)
The a(3) = 12 trees up to relabeling have one of the following 3 forms:
{} {1} {1}
/ \ / \ |
{1} {2,3} {2} {3} {2}
|
{3}
(End)
MAPLE
A005173:=-z*(1+6*z)/(z-1)/(3*z-1)/(2*z-1); # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
CoefficientList[Series[x (1+6 x)/(1-x)/(1-2 x)/(1-3 x), {x, 0, 30}], x] (* Harvey P. Dale, Jul 03 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Feb 06 2001
Name clarified by Andrew Howroyd, Mar 28 2025
STATUS
approved