cp's OEIS Frontend

A180867 Square array read by antidiagonals: T(m,n) is the Wiener index of the Dutch windmill graph D(m,n) (m>=3, n>=1).

%I A180867 #10 Feb 16 2025 08:33:13
%S A180867 3,8,14,15,40,33,27,78,96,60,42,144,189,176,95,64,228,351,348,280,138,
%T A180867 90,352,558,648,555,408,189,125,500,864,1032,1035,810,560,248,165,700,
%U A180867 1230,1600,1650,1512,1113,736,315,216,930,1725,2280,2560,2412,2079,1464
%N A180867 Square array read by antidiagonals: T(m,n) is the Wiener index of the Dutch windmill graph D(m,n) (m>=3, n>=1).
%C A180867 The Dutch windmill graph D(m,n) (also called friendship graph) is the graph obtained by taking n copies of the cycle graph C_m with a vertex in common (i.e., a bouquet of n C_m graphs).
%C A180867 The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
%H A180867 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 A180867 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DutchWindmillGraph.html">Dutch Windmill Graph</a>.
%F A180867 T(3,n) = A033991(n).
%F A180867 T(4,n) = A014642(n).
%F A180867 T(5,n) = A180579(n).
%F A180867 T(6,n) = A180578(n).
%F A180867 T(m,n) = (1/8)n(m^2-1)(2mn-m-2n+2) if m is odd.
%F A180867 T(m,n) = (1/8)n*m^2*(2mn-m-2n+2) if m is even.
%F A180867 The Wiener polynomial of D(m,n) is n*w(m)+(1/2)n(n-1)r(m)^2, where, denoting S(t,p) = Sum_{j=1..p-1}t^j, we have w(m) = mS(t,m/2) + (1/2)mt^(m/2), r(m) = 2S(t,m/2) + t^(m/2) if m is even and w(m) = mS(t,(m+1)/2), r(m) = 2S(t,(m+1)/2) if m is odd.
%e A180867 Square array starts:
%e A180867    3,  14,  33,  60,   95, ...
%e A180867    8,  40,  96, 176,  280, ...
%e A180867   15,  78, 189, 348,  555, ...
%e A180867   27, 144, 351, 648, 1035, ...
%p A180867 T := proc (m, n) if `mod`(m, 2) = 1 then (1/8)*n*(m^2-1)*(2*m*n-m-2*n+2) else (1/8)*n*m^2*(2*m*n-m-2*n+2) end if end proc: for n from 3 to 12 do seq(T(n+1-j, j), j = 1 .. n-2) end do; # yields sequence in triangular form
%Y A180867 Cf. A014642, A033991, A180578, A180579.
%K A180867 nonn,tabl
%O A180867 3,1
%A A180867 _Emeric Deutsch_, Sep 30 2010
%E A180867 Typos in Wiener polynomial information corrected by _Emeric Deutsch_, Sep 30 2010