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.

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

Author

Emeric Deutsch, Sep 16 2010

Keywords

• nonn
• tabf

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).

Examples

			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;
		

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).

Crossrefs

Cf. A060544, A180571.

Programs

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

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)).