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%I #6 Mar 30 2020 04:17:36
%S 3,8,21,15,48,63,27,85,136,138,42,144,235,288,255,64,217,387,490,520,
%T 423,90,320,574,792,875,848,651,125,441,832,1162,1395,1415,1288,948,
%U 165,600,1134,1664,2030,2232,2135,1856,1323,216,781,1525,2250,2880,3227
%N Square array read by antidiagonals: T(m,n) is the Wiener index of the prism graph C(m) x P(n) (m>=3, n>=1). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
%C T(m,2) = A138179(m).
%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/PrismGraph.html">Prism Graph</a>.
%F T(m,n) = (1/24)mn([4m(n^2-1)+3n(m^2-1)] if m is odd.
%F T(m,n) = (1/24)m^2*n[4(n^2-1)+3mn] if m is even.
%F The Wiener polynomial p[n](t) for C(2m+1) x P(n) satisfies the recurrence relation p[n]=p[n-1]+p[1]+(2m+1)*a[n], where a[n]=[t+2*sum(t^j,j=2..m+1)*sum(t^j,j=0..n-2) and p[1]=(2m+1)*sum(t^j,j=1..m).
%F The Wiener polynomial q[n](t) for C(2m) x P(n) satisfies the recurrence relation q[n]=q[n-1]+q[1]+2m*b[n], where b[n]=[t+t^{m+1}+2*sum(t^j,j=2..m)*sum(t^j,j=0..n-2) and q[1]=2m*sum(t^j,j=1..m-1)+m*t^m.
%F The above recurrence relations yield simple expressions for the Wiener polynomials (however, they are too long for being reproduced here).
%e Square array starts:
%e 3,21,63,138,255,...
%e 8,48,136,288,520,...
%e 15,85,235,490,875,...
%e 27,144,387,792,1395,...
%p T := proc (m, n) if `mod`(m, 2) = 1 then (1/24)*m*n*(4*m*(n^2-1)+3*n*(m^2-1)) else (1/24)*m^2*n*(4*n^2-4+3*m*n) end if end proc: for m from 3 to 12 do seq(T(m+1-j, j), j = 1 .. m-2) end do; # yields sequence in triangular form
%Y Cf. A138179.
%K nonn,tabl
%O 3,1
%A _Emeric Deutsch_, Sep 29 2010
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