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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.
1
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>=1} k*T(n,k) = A180576(n) = the Wiener index of the corresponding graph.
LINKS
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 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).
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).
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
Sequence in context: A221819 A329026 A307535 * A115402 A297790 A260462
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 19 2010
STATUS
approved