

A180855


Square array read by antidiagonals: T(m,n) is the Wiener index of the banana tree B(n,k) (n>=1, k>=2). B(n,k) is the graph obtained by taking n copies of a star graph on k nodes and connecting with an edge one leaf of each of these n stars with an additional node.


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4, 20, 10, 48, 56, 18, 88, 138, 108, 28, 140, 256, 270, 176, 40, 204, 410, 504, 444, 260, 54, 280, 600, 810, 832, 660, 360, 70, 368, 826, 1188, 1340, 1240, 918, 476, 88, 468, 1088, 1638, 1968, 2000, 1728, 1218, 608, 108, 580, 1386, 2160, 2716, 2940, 2790
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OFFSET

1,1


COMMENTS

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.


LINKS

Table of n, a(n) for n=1..51.
B. E. Sagan, YN. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959969.
Eric Weisstein's World of Mathematics, Banana Tree.


FORMULA

T(n,k) = n(k1)(3nk2k+2).
T(n,2) = A033579(n).
T(n,4) = A060787(n+2).
The Wiener polynomial of the tree B(n,k) is W(n,k,t)=(1/2)nt(a+bt+ct^2+dt^3+et^4+ft^5), where a=2k, b=3+n+k^23k, c=2n+2k6, d=(n1)(2k3), e=2(n1)(k2), and f=(n1)(k2)^2.


EXAMPLE

T(1,2)=4 because the banana tree B(1,2) reduces to a path on 3 nodes, where the distances are 1, 1, and 2.
Square array T(n,k) begins:
4,10,18,28,40,54,70;
20,56,108,176,260,360,476;
48,138,270,444,660,918,1218;
88,256,504,832,1240,1728,2296;


MAPLE

T := proc (n, k) options operator, arrow: n*(k1)*(3*n*k2*k+2) end proc: for n to 10 do seq(T(n+2j, j), j = 2 .. n+1) end do; # yields sequence in triangular form


CROSSREFS

Cf. A033579, A060787
Sequence in context: A125514 A118392 A263964 * A213822 A182456 A196380
Adjacent sequences: A180852 A180853 A180854 * A180856 A180857 A180858


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Sep 24 2010


STATUS

approved



