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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.

%I A180855 #15 Feb 16 2025 08:33:13
%S A180855 4,20,10,48,56,18,88,138,108,28,140,256,270,176,40,204,410,504,444,
%T A180855 260,54,280,600,810,832,660,360,70,368,826,1188,1340,1240,918,476,88,
%U A180855 468,1088,1638,1968,2000,1728,1218,608,108,580,1386,2160,2716,2940,2790,2296,1560,756,130
%N 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.
%C A180855 The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
%H A180855 B. E. Sagan, Y-N. Yeh, and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%H A180855 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BananaTree.html">Banana Tree</a>.
%F A180855 T(n,k) = n(k-1)(3nk-2k+2).
%F A180855 T(n,2) = A033579(n).
%F A180855 T(n,4) = A060787(n+2).
%F A180855 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.
%e A180855 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.
%e A180855 Square array T(n,k) begins:
%e A180855 4,10,18,28,40,54,70;
%e A180855 20,56,108,176,260,360,476;
%e A180855 48,138,270,444,660,918,1218;
%e A180855 88,256,504,832,1240,1728,2296;
%p A180855 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
%t A180855 T[n_, k_] := n*(k - 1)*(3*n*k - 2*k + 2);
%t A180855 Table[T[n - k + 2, k], {n, 1, 10}, {k, 2, n + 1}] // Flatten (* _Jean-François Alcover_, Aug 26 2024 *)
%Y A180855 Cf. A033579, A060787
%K A180855 nonn,tabl
%O A180855 1,1
%A A180855 _Emeric Deutsch_, Sep 24 2010