A180575 - OEIS

%I #7 Mar 29 2020 13:10:39

%S 12,15,9,16,24,20,6,20,35,35,15,24,42,48,30,9,28,49,63,49,21,32,56,72,

%T 64,40,12,36,63,81,81,63,27,40,70,90,90,80,50,15,44,77,99,99,99,77,33,

%U 48,84,108,108,108,96,60,18,52,91,117,117,117,117,91,39,56,98,126,126

%N Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the prism graph C_n X P_3 with the edges of the outer cycle removed (called a web graph). Equivalently, the graph is obtained by attaching a pendant edge to each node of the outer cycle of the circular ladder (prism) C_n X P_2. C_n denotes the cycle graph on n nodes and P_n denotes the path graph on n nodes.

%C The entries in row n are the coefficients of the Wiener polynomial of the graph.

%C Number of entries in row n is 2+floor(n/2).

%C The entries in row n are the coefficients of the Wiener polynomial of the corresponding graph.

%C Sum of entries in row n is 3n(3n-1)/2 = A062741.

%C Sum(k*T(n,k), k>=1) = A180576(n) = the Wiener index of the corresponding graph.

%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/WebGraph.html">Web Graph</a>

%F The generating polynomial of row 2n+1 (which is also the Wiener polynomial of the corresponding graph) is (2n+1){4t+3t^2+2t^3-2t^{n+1}-4t^{n+2}-3*t^{n+3}]/(1-t).

%F The generating polynomial of row 2n (which is also the Wiener polynomial of the corresponding graph) is n[8t+6t^2+4t^3-2t^n-6t^{n+1}-7t^{n+2}-3t^{n+3}]/(1-t).

%e The triangle starts:

%e 12,15,9;

%e 16,24,20,6;

%e 20,35,35,15;

%e 24,42,48,30,9;

%p P := proc (n) if `mod`(n, 2) = 1 then sort(expand(simplify(n*(4*t+3*t^2+2*t^3-2*t^((1/2)*n+1/2)-4*t^((1/2)*n+3/2)-3*t^((1/2)*n+5/2))/(1-t)))) else sort(expand(simplify((1/2)*n*(8*t+6*t^2+4*t^3-2*t^((1/2)*n)-6*t^((1/2)*n+1)-7*t^((1/2)*n+2)-3*t^((1/2)*n+3))/(1-t)))) end if end proc: for n from 3 to 14 do seq(coeff(P(n), t, j), j = 1 .. 2+floor((1/2)*n)) end do; # yields sequence in triangular form

%Y Cf. A062741, A180576.

%K nonn,tabf

%O 3,1

%A _Emeric Deutsch_, Sep 19 2010