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.

12, 15, 9, 16, 24, 20, 6, 20, 35, 35, 15, 24, 42, 48, 30, 9, 28, 49, 63, 49, 21, 32, 56, 72, 64, 40, 12, 36, 63, 81, 81, 63, 27, 40, 70, 90, 90, 80, 50, 15, 44, 77, 99, 99, 99, 77, 33, 48, 84, 108, 108, 108, 96, 60, 18, 52, 91, 117, 117, 117, 117, 91, 39, 56, 98, 126, 126

Offset

3,1

Comments

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

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

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

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

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

Links

Table of n, a(n) for n=3..69.

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Eric Weisstein's World of Mathematics, Web Graph

Formula

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).

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).

Example

The triangle starts:
12,15,9;
16,24,20,6;
20,35,35,15;
24,42,48,30,9;

Maple

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

Crossrefs

Cf. A062741, A180576.

Sequence in context: A221819 A329026 A307535 * A115402 A297790 A260462

Adjacent sequences: A180572 A180573 A180574 * A180576 A180577 A180578

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 19 2010

Status

approved