OFFSET
0,3
COMMENTS
The binomial trees b(k) of order k are ordered trees defined as follows: 1. b(0) consists of a single node. 2. For k>=1, b(k) is obtained from two copies of b(k-1) by linking them in such a way that the root of one is the leftmost child of the root of the other. See the Iyer & Reddy references.
REFERENCES
K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of Binomial trees and Fibonacci trees, Int'l. J. Math. Engin. with Comp., Accepted for publication, Sept. 2009.
T. H. Cormen, C. E. Leiserson and R. L. Rivest: Introduction to Algorithms. MIT Press / McGraw-Hill (1990)
LINKS
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of binomial trees and Fibonacci trees, arXiv:0910.4432 [cs.DM], 2009.
Index entries for linear recurrences with constant coefficients, signature (10,-32,32).
FORMULA
a(n) = Sum_{k>=1} k*A192020(n,k).
From Colin Barker, Jul 07 2012: (Start)
a(n) = 2^(n-1)*(1+2^n*(n-1)).
a(n) = 10*a(n-1) - 32*a(n-2) + 32*a(n-3).
G.f.: x/((1-2*x)*(1-4*x)^2). (End)
MAPLE
a := proc(n) (n-1)*2^(2*n-1)+2^(n-1) end proc: seq(a(n), n = 0 .. 23);
MATHEMATICA
LinearRecurrence[{10, -32, 32}, {0, 1, 10}, 23] (* Jean-François Alcover, Sep 23 2017 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 22 2011
EXTENSIONS
Initial 0 in the sample values which is Wiener index of singleton tree b(0), and consequent amendments to formulas. - Kevin Ryde, Sep 12 2019
STATUS
approved