A180862 - OEIS

%I #6 Mar 30 2020 04:18:06

%S 4,10,10,18,20,20,28,32,35,35,40,46,52,56,56,54,62,71,79,84,84,70,80,

%T 92,104,114,120,120,88,100,115,131,146,158,165,165,108,122,140,160,

%U 180,198,212,220,220,130,146,167,191,216,240,261,277,286,286,154,172,196,224

%N Square array read by antidiagonals: T(m,n) is the Wiener index of the flower graph F(m,n) (m>=2, n>=1). F(m,n) is the graph obtained by joining with an edge a node in the star graph on m nodes to an end-node of the path graph P_n. The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StarGraph.html">Star Graph</a>.

%F T(m,n) = (m-1)^2+(1/6)n(n^2-19)+(1/2)mn(n+5).

%F G.f.: G(t,s)=Sum(Sum(T(m,n)*t^m*s^n, n=1..infinity),m=2..infinity) = t^2*s(2-t-s)(ts^2-2ts+s^2-2s+2)/[(1-t)^3*(1-s)^4].

%F The Wiener polynomial of the graph F(m,n) is (m-1)t+(1/2)(m-1)(m-2)t^2+t(t^n-nt+n-1)/(1-t)^2+t[1+t+(m-2)t^2](1-t^n)/(1-t).

%e Square array T(i,j) begins:

%e 4, 10, 20, 35, 56, 84, ...

%e 10, 20, 35, 56, 84, 120, ...

%e 18, 32, 52, 79, 114, 158, ...

%e 28, 46, 71, 104, 146, 198, ...

%p T := proc (m, n) options operator, arrow: (m-1)^2+(1/6)*n*(n^2-19)+(1/2)*m*n*(n+5) end proc; for n from 2 to 12 do seq(T(n+1-i, i), i = 1 .. n-1) end do; # yields sequence in triangular form

%K nonn,tabl

%O 2,1

%A _Emeric Deutsch_, Sep 27 2010