A192017 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the Fibonacci tree of order n (1<=k<=n; entries in row n are the coefficients of the corresponding Wiener polynomial).

1, 2, 1, 4, 4, 2, 7, 10, 9, 2, 12, 21, 27, 15, 3, 20, 40, 65, 57, 25, 3, 33, 72, 138, 163, 114, 37, 4, 54, 125, 270, 394, 378, 206, 54, 4, 88, 212, 500, 854, 1033, 796, 354, 74, 5, 143, 354, 891, 1716, 2479, 2463, 1571, 574, 100, 5, 232, 585, 1545, 3265, 5424, 6559, 5469, 2917, 896, 130, 6

Offset

1,2

Comments

The Fibonacci trees f(k) of order k are defined as follows: 1. f(-1) and f(0) each consist of a single node. 2. For k>=1, to the root of f(k-1), taken as the root of f(k), we attach with a rightmost edge the tree f(k-2). See the Iyer & Reddy references. These trees are not the same as the Fibonacci trees in A180566.

Sum of entries in row n is A191797(n+2).

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=1..66.

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

T(n,1) = A000071(n-2) (Fibonacci numbers minus 1).

Sum_{k=1..n} k*T(n,k) = A165910(n) (the Wiener indices).

The Wiener polynomial w(n,t) of the 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)*r(n-2,t), w(0,t)=w(1,t)=0, where r(n,t) is the generating polynomial of the nodes of the Fibonacci tree of order n with respect to the level of the nodes (for example, r(2,t) = 1 + 2t for the tree /\; see A011973 and the Maple program).

Example

T(2,2)=1 because in the Fibonacci tree of order 2, namely /\, there is only 1 pair of nodes at distance 2 (the two leaves).
Triangle starts:
   1;
   2,  1;
   4,  4,  2;
   7, 10,  9,  2;
  12, 21, 27, 15,  3;
  20, 40, 65, 57, 25,  3;

Maple

G := (1+t*z)/(1-z-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 11 do r[n] := sort(coeff(Gser, z, n)) end do; w[0] := 0; w[1] := t; for n from 2 to 11 do w[n] := sort(expand(w[n-1]+w[n-2]+t*r[n-1]*r[n-2])) end do: for n from 1 to 11 do seq(coeff(w[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form

Mathematica

m = 11; Gser = Series[(1 + t*z)/(1 - z - t*z^2), {z, 0, m}]; Do[r[n] = Coefficient[Gser, z, n], {n, 0, m}]; w[0] = 0; w[1] = t; Do[w[n] = Expand[w[n - 1] + w[n - 2] + t*r[n - 1]*r[n - 2]] , {n, 2, m}]; Flatten[Table[Coefficient[w[n], t, j], {n, 1, m}, {j, 1, n}]] (* Jean-François Alcover, Sep 02 2011, after Maple *)

Crossrefs

Cf. A000071, A011973, A165910, A191797.

Sequence in context: A113421 A135366 A247248 * A180566 A051289 A134308

Adjacent sequences: A192014 A192015 A192016 * A192018 A192019 A192020

Keyword

nonn,tabl

Author

Emeric Deutsch, Jun 21 2011

Status

approved