OFFSET
2,1
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.
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.
FORMULA
Generating polynomial of row n is (2q(1-q^n) - n(1-q^2))^2/(2(1-q)^4) - n^2/2.
EXAMPLE
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;
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;
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 19 2008
STATUS
approved