A143944 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k from each other in the grid P_n X P_n (1 <= k <= 2n-2), where P_n is the path graph on n vertices.

4, 2, 12, 14, 8, 2, 24, 34, 32, 20, 8, 2, 40, 62, 68, 60, 40, 20, 8, 2, 60, 98, 116, 116, 100, 70, 40, 20, 8, 2, 84, 142, 176, 188, 180, 154, 112, 70, 40, 20, 8, 2, 112, 194, 248, 276, 280, 262, 224, 168, 112, 70, 40, 20, 8, 2, 144, 254, 332, 380, 400, 394, 364, 312, 240

Offset: 2

Author

Emeric Deutsch, Sep 19 2008

Keywords

• nonn
• tabf

Comments

Row n contains 2n-2 entries.

Sum of entries in row n = n^2*(n^2 - 1)/2 = A083374(n).

The entries in row n are the coefficients of the Wiener (Hosoya) polynomial of the grid P_n X P_n.

Sum_{k=1..2n-2} k*T(n,k) = n^3*(n^2 - 1)/3 = A143945(n) = the Wiener index of the grid P_n X P_n.

The average of all distances in the grid P_n X P_n is 2n/3.

Examples

			T(2,2)=2 because P_2 X P_2 is a square and there are 2 pairs of vertices at distance 2.
Triangle starts:
   4,  2;
  12, 14,  8,  2;
  24, 34, 32, 20,  8,  2;
  40, 62, 68, 60, 40, 20,  8,  2;
		

Links

• D. Stevanovic, Hosoya polynomial of composite graphs, Discrete Math., 235 (2001), 237-244.
• B.-Y. Yang and Y.-N. Yeh, Wiener polynomials of some chemically interesting graphs, International Journal of Quantum Chemistry, 99 (2004), 80-91.
• Y.-N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math., 135 (1994), 359-365.

Crossrefs

Cf. A083374, A143945.

Programs

Maple

for n from 2 to 10 do Q[n]:=sort(expand(simplify((1/2)*(2*q*(1-q^n)-n*(1-q^2))^2/(1-q)^4-(1/2)*n^2))) end do: for n from 2 to 9 do seq(coeff(Q[n],q,j),j= 1..2*n-2) end do;

Formula

Generating polynomial of row n is (2q(1-q^n) - n(1-q^2))^2/(2(1-q)^4) - n^2/2.