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.

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

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, 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, Banana Tree.

Formula

T(n,k) = n(k-1)(3nk-2k+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^2-3k, c=2n+2k-6, d=(n-1)(2k-3), e=2(n-1)(k-2), and f=(n-1)(k-2)^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*(k-1)*(3*n*k-2*k+2) end proc: for n to 10 do seq(T(n+2-j, 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