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A192024 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the double-comb graph \/\/\/...\/_\/ with 3n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.

%I A192024 #10 Mar 30 2020 03:18:36
%S A192024 2,1,5,6,4,8,12,12,4,11,18,21,12,4,14,24,30,21,12,4,17,30,39,30,21,12,
%T A192024 4,20,36,48,39,30,21,12,4,23,42,57,48,39,30,21,12,4,26,48,66,57,48,39,
%U A192024 30,21,12,4,29,54,75,66,57,48,39,30,21,12,4,32,60,84,75,66,57,48,39,30,21,12,4
%N A192024 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the double-comb graph \/_\/_\/...\/_\/ with 3n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.
%C A192024 Row n contains n+1 entries.
%C A192024 Sum of entries in row n is 3*n*(3*n-1)/2=A062741(n).
%C A192024 Sum(k*T(n,k),k>=1)=A192025(n) (the Wiener indices).
%H A192024 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.
%F A192024 G.f.: G(t,z)=t*z*(2+t+z+2*t*z+3*t^2*z)/((1-t*z)*(1-z)^2).
%F A192024 G.f. of column 1: z*(2+z)/(1-z)^2.
%F A192024 G.f. of column 2: z*(1+4*z+z^2)/(1-z)^2.
%F A192024 G.f. of  column k>=3: z^(k-1)*(4+4*z+z^2)/(1-z)^2.
%F A192024 The generating polynomial of row n (i.e. the Wiener polynomial of the double-comb with 3n nodes) is n*(2*t +t^2)+t*(1+2*t)^2*(n*(1-t)-(1-t^n))/(1-t)^2 or, equivalently, n*(2*t+t^2)+t*(1+2*t)^2*Sum((n-j)*t^(j-1), j=1..n-1).
%e A192024 T(2,1)=5, T(2,2)=6, T(2,3)=4 because in the graph \/_\/ there are 5 pairs of nodes at distance 1, 6 pairs at distance 2, and 4 pairs at distance 3.
%e A192024 Triangle starts:
%e A192024 2,1;
%e A192024 5,6,4;
%e A192024 8,12,12,4;
%e A192024 11,18,21,12,4;
%p A192024 Q := proc (n) options operator, arrow: n*(t^2+2*t)+t*(1+2*t)^2*(sum((n-j)*t^(j-1), j = 1 .. n-1)) end proc: for n to 11 do P[n] := sort(expand(Q(n))) end do: for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form
%Y A192024 Cf. A062741, A192025
%K A192024 nonn,tabf
%O A192024 1,1
%A A192024 _Emeric Deutsch_, Jun 25 2011