

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



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Sum of entries in row n = n(2n1) = 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.


LINKS



FORMULA

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


EXAMPLE

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;


MAPLE

G:=q*z*(1+2*z+q*z)/((1z)^2*(1q*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


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



