A143370 - OEIS

%I #13 Jul 21 2017 10:47:00

%S 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,

%T 2,22,26,22,18,14,10,6,2,25,30,26,22,18,14,10,6,2,28,34,30,26,22,18,

%U 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

%N 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.

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

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

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

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

%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.

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

%e T(2,1)=4 because in the graph P_2 x P_2 (a square) we have 4 distances equal to 1.

%e Triangle starts:

%e    1;

%e    4,  2;

%e    7,  6,  2;

%e   10, 10,  6,  2;

%e   13, 14, 10,  6,  2;

%p 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

%Y Cf. A000384.

%K nonn,tabl

%O 1,2

%A _Emeric Deutsch_, Sep 05 2008