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A180572 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the circular ladder P_2 X C_n (also called a prism), where P_2 is the path graph on 2 nodes and C_n is the cycle graph on n nodes.

%I A180572 #14 Nov 08 2017 05:11:09
%S A180572 9,6,12,12,4,15,20,10,18,24,18,6,21,28,28,14,24,32,32,24,8,27,36,36,
%T A180572 36,18,30,40,40,40,30,10,33,44,44,44,44,22,36,48,48,48,48,36,12,39,52,
%U A180572 52,52,52,52,26,42,56,56,56,56,56,42,14,45,60,60,60,60,60,60,30,48,64,64
%N A180572 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the circular ladder P_2 X C_n (also called a prism), where P_2 is the path graph on 2 nodes and C_n is the cycle graph on n nodes.
%C A180572 Row n contains 1 + floor(n/2) entries.
%C A180572 Sum of entries in row n = n(2n-1) = A000384(n).
%C A180572 T(n,1) = 3n = number of edges in the corresponding graph.
%C A180572 Sum_{k>=1} k*T(n,k) = A138179(n).
%C A180572 The generating polynomial of row n (i.e., the Wiener polynomial of the circular ladder of order n) has been obtained from the Wiener polynomial of the cycle C_n (see the Sagan et al. paper) and by determining the distribution of the distances from the nodes of one cycle to the nodes of the other cycle. They can also be derived from the Doslic paper (Corollary 11 and Lemma 1).
%D A180572 J. Gross and J. Yellen, Graph Theory and its Applications, CRC, Boca Raton, 1999 (p. 14).
%H A180572 T. Doslic, <a href="http://amc-journal.eu/index.php/amc/article/view/15">Vertex-weighted Wiener polynomials for composite graphs</a>, Ars Mathematica Contemporanea, 1, 2008, 66-80.
%H A180572 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 A180572 The generating polynomial for row 2n+1 is (2n+1)(3t+t^2-2t^{n+1}-2t^{n+2})/(1-t) and for row 2n it is 2n(3t+t^2-t^n-2t^{n+1}-t^{n+2})/(1-t) (these are also the Wiener polynomials of the corresponding circular ladders).
%F A180572 The bivariate g.f. G=G(t,z) appears in the Maple program.
%e A180572 T(3,2)=6 because in P_2 X C_3 there are six unordered pairs of nodes at distance 2 (from the vertices of the outer triangle to the "opposite" vertices of the inner triangle).
%e A180572 Triangle starts:
%e A180572    9,  6;
%e A180572   12, 12,  4;
%e A180572   15, 20, 10;
%e A180572   18, 24, 18,  6;
%e A180572   21, 28, 28, 14;
%p A180572 G := t*z^3*(9+6*t-6*z+4*t^2*z-16*t*z^2-10*t^2*z^2+8*t*z^3 +2*t^2*z^3 -2*t^3*z^3 +7*t^2*z^4+4*t^3*z^4-4*t^2*z^5-2*t^3*z^5) / ((1-z)^2*(1-t*z^2)^2): Gser := simplify(series(G, z = 0, 19)): for n from 3 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 3 to 16 do seq(coeff(P[n], t, j), j = 1 .. 1+floor((1/2)*n)) end do; # yields sequence in triangular form
%Y A180572 Cf. A000384, A138179.
%K A180572 nonn,tabf
%O A180572 3,1
%A A180572 _Emeric Deutsch_, Sep 16 2010