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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
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, 421-427 (see Eq. between (10) and (11); replace n with n+2).
FORMULA
The generating polynomial of row n is t*(9t^(n+2) - 3nt^3 - 8t^2 - 2t + 1 + 3n)/(1-t)^2.
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)).
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^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
CROSSREFS
Sequence in context: A152199 A293926 A038598 * A074474 A070420 A223423
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 16 2010
STATUS
approved