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
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
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 19 2010
STATUS
approved