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%I #22 Dec 10 2016 03:00:24
%S 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,
%T 114,37,4,54,125,270,394,378,206,54,4,88,212,500,854,1033,796,354,74,
%U 5,143,354,891,1716,2479,2463,1571,574,100,5,232,585,1545,3265,5424,6559,5469,2917,896,130,6
%N 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).
%C 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.
%C Sum of entries in row n is A191797(n+2).
%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<959::AID-QUA2>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 T(n,1) = A000071(n-2) (Fibonacci numbers minus 1).
%F Sum_{k=1..n} k*T(n,k) = A165910(n) (the Wiener indices).
%F 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).
%e 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).
%e Triangle starts:
%e 1;
%e 2, 1;
%e 4, 4, 2;
%e 7, 10, 9, 2;
%e 12, 21, 27, 15, 3;
%e 20, 40, 65, 57, 25, 3;
%p 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
%t 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 *)
%Y Cf. A000071, A011973, A165910, A191797.
%K nonn,tabl
%O 1,2
%A _Emeric Deutsch_, Jun 21 2011
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