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A180566 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 (entries in row n are the coefficients of the corresponding Wiener polynomial).

%I A180566 #7 Jan 05 2025 19:51:39
%S A180566 2,1,4,4,2,8,10,8,6,4,14,19,20,18,18,12,4,24,34,40,44,48,46,40,20,4,
%T A180566 40,58,72,88,106,114,122,112,76,28,4,66,97,124,160,208,242,284,310,
%U A180566 308,244,128,36,4,108,160,208,276,376,466,576,686,782,812,720,472,196,44,4,176
%N A180566 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 (entries in row n are the coefficients of the corresponding Wiener polynomial).
%C A180566 A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.
%C A180566 Row n (n>=3) contains 2n-3 entries.
%C A180566 Sum of entries of row n is (F(n+1) - 1)(2F(n+1) - 1), where F(j)=A000045(j) are the Fibonacci numbers.
%C A180566 T(n,1) = 2*F(n+1)-2 = number of edges in the Fibonacci tree of order n.
%C A180566 Sum(k*T(n,k), k>=0) = A180567 (the Wiener index of the Fibonacci tree of order n).
%D A180566 D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
%H A180566 Y. Horibe, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
%H A180566 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.
%F A180566 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) + t*r(n-2,t) + t^2*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, 1+2t for the tree /\; see A178522 and the Maple program).
%e A180566 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 A180566 Triangle starts:
%e A180566 2,1;
%e A180566 4,4,2;
%e A180566 8,10,8,6,4;
%e A180566 14,19,20,18,18,12,4;
%p A180566 G := (1-t*z+t*z^2)/((1-z)*(1-t*z-t*z^2)): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 20 do r[n] := sort(coeff(Gser, z, n)) end do: w[0] := 0: w[1] := 0: for n from 2 to 15 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: 2, 1; for n from 3 to 10 do seq(coeff(w[n], t, j), j = 1 .. 2*n-3) end do; # yields sequence in triangular form
%Y A180566 Cf. A178522, A180567
%K A180566 nonn,tabf
%O A180566 2,1
%A A180566 _Emeric Deutsch_, Sep 14 2010