A192018 - OEIS

%I #11 Dec 10 2016 03:00:16

%S 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,

%T 56,66,74,78,77,61,32,9,1,53,71,98,124,152,175,196,203,180,118,49,11,

%U 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

%N 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<=2n-3; entries in row n are the coefficients of the corresponding Wiener polynomial).

%C 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(k-1) as its left subtree and f(k-2) as its right subtree. See the Iyer & Reddy references.

%C Row n contains 2n-3 entries.

%C T(n,1) = A001911(n-1) (Fibonacci numbers minus 2).

%C Sum_{k>=1} k*T(n,k) = A192019(n) (the Wiener indices).

%D 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.

%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1996)60:5&lt;959::AID-QUA2&gt;3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.

%H K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, <a href="http://arxiv.org/abs/0910.4432">Wiener index of binomial trees and Fibonacci trees</a>, arXiv:0910.4432 [cs.DM], 2009.

%F The Wiener polynomial w(n,t) of the binary Fibonacci tree of order n satisfies the recurrence relation w(n,t) = w(n-1,t) + w(n-2,t) + t*r(n-1,t) + t*r(n-2) + t^2*r(n-1,t)*r(n-2,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).

%e Triangle starts:

%e    1;

%e    3,  2,  1;

%e    6,  6,  5,  3,  1;

%e   11, 13, 14, 12, 10,  5,  1;

%e   19, 24, 30, 31, 31, 28, 19,  7,  1;

%p G := z/((1-z)*(1-t*z-t*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[n-1]+w[n-2]+t*r[n-1]+t*r[n-2]+t^2*r[n-1]*r[n-2])) end do: for n from 2 to 10 do seq(coeff(w[n], t, k), k = 1 .. 2*n-3) end do; # yields sequence in triangular form

%Y Cf. A001911, A192019.

%K nonn,tabf

%O 2,2

%A _Emeric Deutsch_, Jun 21 2011