This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A180575 #13 Feb 16 2025 08:33:13
%S A180575 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 A180575 64,40,12,36,63,81,81,63,27,40,70,90,90,80,50,15,44,77,99,99,99,77,33,
%U A180575 48,84,108,108,108,96,60,18,52,91,117,117,117,117,91,39,56,98,126,126
%N A180575 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 A180575 The entries in row n are the coefficients of the Wiener polynomial of the graph.
%C A180575 Number of entries in row n is 2+floor(n/2).
%C A180575 The entries in row n are the coefficients of the Wiener polynomial of the corresponding graph.
%C A180575 Sum of entries in row n is 3n(3n-1)/2 = A062741.
%C A180575 Sum_{k>=1} k*T(n,k) = A180576(n) = the Wiener index of the corresponding graph.
%H A180575 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 A180575 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WebGraph.html">Web Graph</a>
%F A180575 The generating polynomial of row 2*n+1 (which is also the Wiener polynomial of the corresponding graph) is (2*n+1)*(4*t+3*t^2+2*t^3-2*t^(n+1)-4*t^(n+2)-3*t^(n+3))/(1-t).
%F A180575 The generating polynomial of row 2*n (which is also the Wiener polynomial of the corresponding graph) is n*(8*t+6*t^2+4*t^3-2*t^n-6*t^(n+1)-7*t^(n+2)-3*t^(n+3))/(1-t).
%e A180575 The triangle starts:
%e A180575 12,15,9;
%e A180575 16,24,20,6;
%e A180575 20,35,35,15;
%e A180575 24,42,48,30,9;
%p A180575 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 A180575 Cf. A062741, A180576.
%K A180575 nonn,tabf
%O A180575 3,1
%A A180575 _Emeric Deutsch_, Sep 19 2010