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%I #5 Dec 16 2016 03:04:17
%S 7,12,9,10,18,18,9,13,24,27,18,9,16,30,36,27,18,9,19,36,45,36,27,18,9,
%T 22,42,54,45,36,27,18,9,25,48,63,54,45,36,27,18,9,28,54,72,63,54,45,
%U 36,27,18,9,31,60,81,72,63,54,45,36,27,18,9,34,66,90,81,72,63,54,45,36,27
%N Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the graph \|/_\/_\/_..._\/_\|/ having n nodes on the horizontal path. The entries in row n are the coefficients of the Wiener polynomial of the graph.
%C Row n has n+1 entries.
%C Sum of entries in row n = (2 + 9n + 9n^2)/2 =A060544(n+1).
%C Sum_{k>=0} k*T(n,k) = A180571(n) (the Wiener indices of the graphs).
%D I. Gutman, SL Lee, CH Chu. YLLuo, Indian J. Chem., 33A, 603.
%D I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte fur Chemie, 131, 421-427 (see Eq. between (10) and (11); replace n with n+2).
%F The generating polynomial of row n is t*(9t^(n+2) - 3nt^3 - 8t^2 - 2t + 1 + 3n)/(1-t)^2.
%F The bivariate g.f. is G = tz^2*(7 + 12t + 9t^2 - 4z - 13tz + 4tz^2 + 6t^2*z^2 - 12t^2*z)/((1-z)^2*(1-tz)).
%e T(2,3)=9 because in the graph \|/_\|/ there are 9 unordered pairs of vertices at distance 3.
%e Triangle starts:
%e 7, 12, 9;
%e 10, 18, 18, 9;
%e 13, 24, 27, 18, 9;
%e 16, 30, 36, 27, 18, 9;
%p for n from 2 to 11 do P[n] := sort(expand(simplify(t*(9*t^(n+2)-3*n*t^3-8*t^2-2*t+1+3*n)/(1-t)^2))) end do: for n from 2 to 11 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form
%Y Cf. A060544, A180571.
%K nonn,tabf
%O 2,1
%A _Emeric Deutsch_, Sep 16 2010
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