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

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

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

Row n contains 2n-3 entries.

T(n,1) = A001911(n-1) (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, 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.

Formula

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).

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/((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

Crossrefs

Cf. A001911, A192019.

Sequence in context: A365743 A208520 A114155 * A079513 A060408 A267121

Adjacent sequences: A192015 A192016 A192017 * A192019 A192020 A192021

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 21 2011

Status

approved