

A192030


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 p1 edges, add a vertex to their union, and connect this vertex with the roots of the star trees.


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1, 4, 4, 9, 20, 9, 16, 48, 48, 16, 25, 88, 117, 88, 25, 36, 140, 216, 216, 140, 36, 49, 204, 345, 400, 345, 204, 49, 64, 280, 504, 640, 640, 504, 280, 64, 81, 368, 693, 936, 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
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OFFSET

1,2


COMMENTS

W(n,1)=W(1,n)=n^2=A000290(n).
W(n,2)=W(2,n)=A033579(p)=2*n*(3*n1).
W(p,n)=W(n,p).


REFERENCES

Stephan G. Wagner, A class of trees and its Wiener index, Acta Appl. Mathem. 91 (2) (2006) 119132.
B. E. Sagan, YN. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959969.


LINKS

Table of n, a(n) for n=1..66.


FORMULA

W(n,p)=n*p*(2*n*pnp+1).
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^2p1), c=n*(n1)*(p1), d=(1/2)*n*(n1)*(p1)^2.


EXAMPLE

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.
The square array starts:
1,4,9,16,25,36,49,...;
4,20,48,88,140,204,280,...;
9,48,117,216,345,504,693,...;
16,88,216,400,640,936,1288,...;


MAPLE

W := proc (n, p) options operator, arrow; n*p*(2*n*pnp+1) end proc: for n to 11 do seq(W(ni, i+1), i = 0 .. n1) end do; # yields sequence in triangular form
W := proc (n, p) options operator, arrow; n*p*(2*n*pnp+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


CROSSREFS

Cf. A000290, A033579
Sequence in context: A135065 A067553 A112683 * A117879 A069549 A118069
Adjacent sequences: A192027 A192028 A192029 * A192031 A192032 A192033


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Jun 29 2011


STATUS

approved



