

A180570


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.


1



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, 22, 42, 54, 45, 36, 27, 18, 9, 25, 48, 63, 54, 45, 36, 27, 18, 9, 28, 54, 72, 63, 54, 45, 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
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OFFSET

2,1


COMMENTS

Row n has n+1 entries.
Sum of entries in row n = (2 + 9n + 9n^2)/2 =A060544(n+1).
Sum_{k>=0} k*T(n,k) = A180571(n) (the Wiener indices of the graphs).


REFERENCES

I. Gutman, SL Lee, CH Chu. YLLuo, Indian J. Chem., 33A, 603.
I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte fur Chemie, 131, 421427 (see Eq. between (10) and (11); replace n with n+2).


LINKS



FORMULA

The generating polynomial of row n is t*(9t^(n+2)  3nt^3  8t^2  2t + 1 + 3n)/(1t)^2.
The bivariate g.f. is G = tz^2*(7 + 12t + 9t^2  4z  13tz + 4tz^2 + 6t^2*z^2  12t^2*z)/((1z)^2*(1tz)).


EXAMPLE

T(2,3)=9 because in the graph \/_\/ there are 9 unordered pairs of vertices at distance 3.
Triangle starts:
7, 12, 9;
10, 18, 18, 9;
13, 24, 27, 18, 9;
16, 30, 36, 27, 18, 9;


MAPLE

for n from 2 to 11 do P[n] := sort(expand(simplify(t*(9*t^(n+2)3*n*t^38*t^22*t+1+3*n)/(1t)^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


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



