

A192018


Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the binary Fibonacci tree of order n (1<=k<=2n3; entries in row n are the coefficients of the corresponding Wiener polynomial).


1



1, 3, 2, 1, 6, 6, 5, 3, 1, 11, 13, 14, 12, 10, 5, 1, 19, 24, 30, 31, 31, 28, 19, 7, 1, 32, 42, 56, 66, 74, 78, 77, 61, 32, 9, 1, 53, 71, 98, 124, 152, 175, 196, 203, 180, 118, 49, 11, 1, 87, 118, 166, 218, 284, 349, 419, 485, 525, 502, 384, 207, 70, 13, 1, 142, 194, 276, 370, 499, 645, 812, 998, 1189, 1331, 1349, 1152, 749, 336, 95, 15, 1
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OFFSET

2,2


COMMENTS

The binary Fibonacci trees f(k) of order k is a rooted binary tree defined as follows: 1. f(0) has no nodes and f(1) consists of a single node. 2. For k>=2, f(k) is constructed from a root with f(k1) as its left subtree and f(k2) as its right subtree. See the Iyer & Reddy references.
Row n contains 2n3 entries.
T(n,1) = A001911(n1) (Fibonacci numbers minus 2).
Sum_{k>=1} k*T(n,k) = A192019(n) (the Wiener indices).


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.


LINKS

Table of n, a(n) for n=2..82.
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.


FORMULA

The Wiener polynomial w(n,t) of the binary Fibonacci tree of order n satisfies the recurrence relation w(n,t) = w(n1,t) + w(n2,t) + t*r(n1,t) + t*r(n2) + t^2*r(n1,t)*r(n2,t), w(1,t)=0, w(2,t)=t, where r(n,t) is the generating polynomial of the nodes of the binary Fibonacci tree f(n) with respect to the level of the nodes (for example, r(2,t) = 1 + t for the tree / ; see A004070 and the Maple program).


EXAMPLE

Triangle starts:
1;
3, 2, 1;
6, 6, 5, 3, 1;
11, 13, 14, 12, 10, 5, 1;
19, 24, 30, 31, 31, 28, 19, 7, 1;


MAPLE

G := z/((1z)*(1t*zt*z^2)): Gser := simplify(series(G, z = 0, 13)): for n to 10 do r[n] := sort(coeff(Gser, z, n)) end do; w[1] := 0: w[2] := t: for n from 3 to 10 do w[n] := sort(expand(w[n1]+w[n2]+t*r[n1]+t*r[n2]+t^2*r[n1]*r[n2])) end do: for n from 2 to 10 do seq(coeff(w[n], t, k), k = 1 .. 2*n3) end do; # yields sequence in triangular form


CROSSREFS

Cf. A001911, A192019.
Sequence in context: A177977 A208520 A114155 * A079513 A060408 A267121
Adjacent sequences: A192015 A192016 A192017 * A192019 A192020 A192021


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Jun 21 2011


STATUS

approved



