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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified January 27 08:51 EST 2022. Contains 350607 sequences. (Running on oeis4.)