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A180854
Square array read by antidiagonals: T(m,n) is the Wiener index of the lollipop graph L(m,n) (m>=1, n>=1). L(m,n) is the graph obtained by joining the complete graph K_m to the path graph P_n by an edge. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
0
1, 4, 4, 8, 10, 10, 13, 17, 20, 20, 19, 25, 31, 35, 35, 26, 34, 43, 51, 56, 56, 34, 44, 56, 68, 78, 84, 84, 43, 55, 70, 86, 101, 113, 120, 120, 53, 67, 85, 105, 125, 143, 157, 165, 165, 64, 80, 101, 125, 150, 174, 195, 211, 220, 220, 76, 94, 118, 146, 176, 206, 234, 258
OFFSET
1,2
COMMENTS
The Wiener polynomial of the graph L(m,n) is (1/2)m(m-1)t + t[t^{n+1}-(n+1)t+n]/(1-t)^2 + (m-1)t^2(1-t^n)/(1-t).
LINKS
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Eric Weisstein's World of Mathematics, Lollipop Graph.
FORMULA
T(m,n) = (1/6)n(n^2-7)+(1/2)m(m+n^2+3n-1).
EXAMPLE
Square array T(i,j) begins:
1 4 10 20 35 56 84 ...
4 10 20 35 56 84 120 ...
8 17 31 51 78 113 157 ...
13 25 43 68 101 143 195 ...
MAPLE
T := proc (m, n) options operator, arrow: (1/6)*n*(n^2-7)+(1/2)*m*(m+n^2+3*n-1) end proc: for m to 11 do seq(T(m+1-j, j), j = 1 .. m) end do; # yields sequence in triangular form
CROSSREFS
Sequence in context: A093340 A357751 A046558 * A014687 A172022 A152967
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Sep 22 2010
STATUS
approved