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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, 2296, 1560, 756, 130
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
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
MATHEMATICA
T[n_, k_] := n*(k - 1)*(3*n*k - 2*k + 2);
Table[T[n - k + 2, k], {n, 1, 10}, {k, 2, n + 1}] // Flatten (* Jean-François Alcover, Aug 26 2024 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Sep 24 2010
STATUS
approved