

A192021


The Wiener index of the binomial tree of order n.


1



0, 1, 10, 68, 392, 2064, 10272, 49216, 229504, 1048832, 4719104, 20972544, 92276736, 402657280, 1744838656, 7516209152, 32212287488, 137439019008, 584115683328, 2473901424640, 10445360988160, 43980466159616, 184717955563520, 774056190148608
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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(k1) 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 / McGrawHill (1990)


LINKS

Table of n, a(n) for n=0..23.
B. E. Sagan, YN. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959969.
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^(n1)*(1+2^n*(n1)).
a(n) = 10*a(n1)  32*a(n2) + 32*a(n3).
G.f.: x/((12*x)*(14*x)^2). (End)


MAPLE

a := proc(n) (n1)*2^(2*n1)+2^(n1) end proc: seq(a(n), n = 0 .. 23);


MATHEMATICA

LinearRecurrence[{10, 32, 32}, {0, 1, 10}, 23] (* JeanFrançois Alcover, Sep 23 2017 *)


CROSSREFS

Cf. A192020.
Sequence in context: A197751 A144052 A280438 * A026984 A104598 A026901
Adjacent sequences: A192018 A192019 A192020 * A192022 A192023 A192024


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



