A192030 - OEIS

%I #8 Mar 30 2020 08:32:59

%S 1,4,4,9,20,9,16,48,48,16,25,88,117,88,25,36,140,216,216,140,36,49,

%T 204,345,400,345,204,49,64,280,504,640,640,504,280,64,81,368,693,936,

%U 1025,936,693,368,81,100,468,912,1288,1500,1500,1288,912,468,100,121,580,1161,1696,2065,2196,2065,1696,1161,580,121

%N Square array read by antidiagonals: W(n,p) (n>=1, p>=1) is the Wiener index of the graph G(n,p) obtained in the following way: consider n copies of a star tree with p-1 edges, add a vertex to their union, and connect this vertex with the roots of the star trees.

%C W(n,1)=W(1,n)=n^2=A000290(n).

%C W(n,2)=W(2,n)=A033579(p)=2*n*(3*n-1).

%C W(p,n)=W(n,p).

%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.

%H Stephan Wagner, <a href="http://dx.doi.org/10.1007/s10440-006-9026-5">A class of trees and its Wiener index</a>, Acta Applic. Mathem. 91 (2) (2006) 119-132.

%F W(n,p)=n*p*(2*n*p-n-p+1).

%F The Wiener polynomial of the graph G(n,p) is a*t+b*t^2+c*t^3+d*t^4, where a=n*p, b=(1/2)*n*(n+p^2-p-1), c=n*(n-1)*(p-1), d=(1/2)*n*(n-1)*(p-1)^2.

%e W(2,2)=20 because G(2,2) is the path graph with 4 edges; its Wiener index is 4*1+3*2+2*3+1*4=20.

%e The square array starts:

%e 1,4,9,16,25,36,49,...;

%e 4,20,48,88,140,204,280,...;

%e 9,48,117,216,345,504,693,...;

%e 16,88,216,400,640,936,1288,...;

%p W := proc (n, p) options operator, arrow; n*p*(2*n*p-n-p+1) end proc: for n to 11 do seq(W(n-i, i+1), i = 0 .. n-1) end do; # yields sequence in triangular form

%p W := proc (n, p) options operator, arrow; n*p*(2*n*p-n-p+1) end proc: for n to 7 do seq(W(n, p), p = 1 .. 10) end do; # yields the first 10 entries in each of the first 7 rows

%Y Cf. A000290, A033579

%K nonn,tabl

%O 1,2

%A _Emeric Deutsch_, Jun 29 2011