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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 p-1 edges, add a vertex to their union, and connect this vertex with the roots of the star trees. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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*n-1).

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

REFERENCES

Stephan G. Wagner, A class of trees and its Wiener index, Acta Appl. Mathem. 91 (2) (2006) 119-132.

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

LINKS

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

FORMULA

W(n,p)=n*p*(2*n*p-n-p+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^2-p-1), c=n*(n-1)*(p-1), d=(1/2)*n*(n-1)*(p-1)^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*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

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

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

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Last modified December 13 02:44 EST 2017. Contains 295954 sequences.