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

2,1

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).

Links

Table of n, a(n) for n=2..74.

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

Cf. A060544, A180571.

Sequence in context: A152199 A293926 A038598 * A074474 A070420 A223423

Adjacent sequences: A180567 A180568 A180569 * A180571 A180572 A180573

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 16 2010

Status

approved