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A180573 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the sun graph on 2n nodes. The sun graph on 2n nodes is obtained by attaching n pendant edges to the cycle graph on n nodes.

%I A180573 #6 Mar 30 2020 05:45:12
%S A180573 6,6,3,8,10,8,2,10,15,15,5,12,18,21,12,3,14,21,28,21,7,16,24,32,28,16,
%T A180573 4,18,27,36,36,27,9,20,30,40,40,35,20,5,22,33,44,44,44,33,11,24,36,48,
%U A180573 48,48,42,24,6,26,39,52,52,52,52,39,13,28,42,56,56,56,56,49,28,7,30,45,60
%N A180573 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the sun graph on 2n nodes. The sun graph on 2n nodes is obtained by attaching n pendant edges to the cycle graph on n nodes.
%C A180573 Number of entries in row n = 2 + floor(n/2).
%C A180573 Sum of entries in row n = n(2n-1)=A000384(n).
%C A180573 Sum(k*T(n,k),k>=1) = A180574(n).
%H A180573 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 A180573 D. Stevanovic, <a href="https://doi.org/10.1016/S0012-365X(00)00277-6">Hosoya polynomial of composite graphs</a>, Discrete Math., 235 (2001), 237-244.
%F A180573 The generating polynomial of row 2n is 2nt-nt^n*(1+t)^2+2nt(1+t)^2*sum(t^j, j=0..n-1); the generating polynomial of row 2n+1 is (2n+1)t[1+(1+t)^2*sum(t^j,j=0..n-1)]; these are the Wiener polynomials of the corresponding graphs.
%e A180573 Triangle starts:
%e A180573 6,6,3;
%e A180573 8,10,8,2;
%e A180573 10,15,15,5;
%e A180573 12,18,21,12,3;
%p A180573 P := proc (n) if `mod`(n, 2) = 0 then sort(expand(n*t*(1+t)^2*(sum(t^j, j = 0 .. (1/2)*n-1))+n*t-(1/2)*n*t^((1/2)*n)*(1+t)^2)) else sort(expand(n*t*(1+(1+t)^2*(sum(t^j, j = 0 .. (1/2)*n-3/2))))) end if end proc; for n from 3 to 15 do P(n) end do: for n from 3 to 15 do seq(coeff(P(n), t, i), i = 1 .. 2+floor((1/2)*n)) end do; # yields sequence in trianguklar form
%Y A180573 Cf. A000384, A180574
%K A180573 nonn,tabf
%O A180573 3,1
%A A180573 _Emeric Deutsch_, Sep 19 2010