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A214581 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the circumcoronene H(n) (n=1,2,3,4,5; see definition in the Klavzar papers).

%I A214581 #13 Aug 06 2024 10:38:40
%S A214581 6,6,3,30,48,57,54,45,30,12,72,126,165,186,195,186,168,138,102,66,27,
%T A214581 132,240,327,390,435,456,462,444,414,366,309,246,177,114,48,210,390,
%U A214581 543,666,765,834,882,900,900,870,825,756,675,582,480,378,270,174,75
%N A214581 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the circumcoronene H(n) (n=1,2,3,4,5; see definition in the Klavzar papers).
%C A214581 The entries in row n are the coefficients of the Wiener polynomial of the corresponding graph.
%C A214581 Row n contains 4n-1 entries.
%C A214581 T(n,1) = 9n^2-3n = A152743(n).
%C A214581 T(n,2) = 6n(3n-2)= A153796(n).
%C A214581 T(n,3) = 3(9n^2-9n+1)= 3*A069131(n) (for n>5 this is a conjecture).
%C A214581 T(n,2n) = n(7n^2-1) = 6*A004126(n) (for n>5 this is a conjecture).
%C A214581 T(n,4n-2) = 6(n^2+n-1) = 6*A028387(n-1) (for n>5 this is a conjecture).
%C A214581 T(n,4n-1) = 3n^2 = A033428(n) (for n>5 this is a conjecture).
%C A214581 Sum(k*T(n,k), k>=1) = A143366(n).
%H A214581 S. Klavzar, <a href="https://citeseerx.ist.psu.edu/pdf/17c4b0fe4eaf5d96053952a0ffb5ac0b1fbfbe3d">A bird's eye view of the cut method and a survey of its applications in chemical graph theory</a>, MATCH, Commun. Math. Comput. Chem. 60, 2008, 255-274.
%H A214581 Bo-Yin Yang and Yeong-Nan Yeh, <a href="https://doi.org/10.1111/j.0022-2526.2004.01516.x">A Crowning Moment for Wiener Indices</a>, Studies in Appl. Math., 112 (2004), 333-340.
%H A214581 Bo-Yin Yang and Yeong-Nan Yeh, <a href="http://www.iis.sinica.edu.tw/papers/byyang/2387-F.pdf">Wiener polynomials of some chemically interesting graphs</a>, International Journal of Quantum Chemistry, 99 (2004), 80-91, 2004.
%H A214581 P. Zigert, S. Klavzar, and I. Gutman, <a href="https://www.researchgate.net/publication/289452341_Calculating_the_hyper-Wiener_index_of_benzenoid_hydrocarbons">Calculating the hyper-Wiener index of benzenoid hydrocarbons</a>, ACH Models Chem., 137, 2000, 83-94.
%F A214581 The entries have been obtained by using the Maple Graph Theory package for finding the distance matrix of each of the five graphs H(n) (n=1,2,3,4,5). The given Maple program yields the Wiener polynomial of H(2) (having as coefficients the entries in row 2).
%Y A214581 Cf. A143366, A214580, A152743, A153796, A069131, A004126, A028387, A033428.
%K A214581 nonn,tabf,more
%O A214581 1,1
%A A214581 _Emeric Deutsch_, Aug 31 2012