A143370 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the grid P_2 x P_n (1 <= k <= n). P_m is the path graph on m vertices.

1, 4, 2, 7, 6, 2, 10, 10, 6, 2, 13, 14, 10, 6, 2, 16, 18, 14, 10, 6, 2, 19, 22, 18, 14, 10, 6, 2, 22, 26, 22, 18, 14, 10, 6, 2, 25, 30, 26, 22, 18, 14, 10, 6, 2, 28, 34, 30, 26, 22, 18, 14, 10, 6, 2, 31, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2, 34, 42, 38, 34, 30, 26, 22, 18, 14, 10, 6, 2

Offset: 1

Author

Emeric Deutsch, Sep 05 2008

Keywords

• nonn
• tabl

Comments

Sum of entries in row n = n(2n-1) = A000384(n).

The entries in row n are the coefficients of the Wiener polynomial of the grid P_2 x P_n.

Sum_{k=1..n} k*T(n,k) = A131423(n) = the Wiener index of the grid P_2 x P_n.

The average of all distances in the grid P_2 x P_n is (n+2)/3.

Examples

			T(2,1)=4 because in the graph P_2 x P_2 (a square) we have 4 distances equal to 1.
Triangle starts:
   1;
   4,  2;
   7,  6,  2;
  10, 10,  6,  2;
  13, 14, 10,  6,  2;
		

Links

• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Crossrefs

Cf. A000384.

Programs

Maple

G:=q*z*(1+2*z+q*z)/((1-z)^2*(1-q*z)): Gser:= simplify(series(G,z=0,15)): for n to 12 do p[n]:=sort(coeff(Gser,z,n)) end do: for n to 12 do seq(coeff(p[n],q, j),j=1..n) end do; # yields sequence in triangular form

Formula

G.f. = G(q,z) = qz(1+2z+qz)/((1-qz)(1-z)^2).