OFFSET
1,1
COMMENTS
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.
FORMULA
G.f.: G(t,z)=t*z*(2+t+z+2*t*z+3*t^2*z)/((1-t*z)*(1-z)^2).
G.f. of column 1: z*(2+z)/(1-z)^2.
G.f. of column 2: z*(1+4*z+z^2)/(1-z)^2.
G.f. of column k>=3: z^(k-1)*(4+4*z+z^2)/(1-z)^2.
The generating polynomial of row n (i.e. the Wiener polynomial of the double-comb with 3n nodes) is n*(2*t +t^2)+t*(1+2*t)^2*(n*(1-t)-(1-t^n))/(1-t)^2 or, equivalently, n*(2*t+t^2)+t*(1+2*t)^2*Sum((n-j)*t^(j-1), j=1..n-1).
EXAMPLE
T(2,1)=5, T(2,2)=6, T(2,3)=4 because in the graph \/_\/ there are 5 pairs of nodes at distance 1, 6 pairs at distance 2, and 4 pairs at distance 3.
Triangle starts:
2,1;
5,6,4;
8,12,12,4;
11,18,21,12,4;
MAPLE
Q := proc (n) options operator, arrow: n*(t^2+2*t)+t*(1+2*t)^2*(sum((n-j)*t^(j-1), j = 1 .. n-1)) end proc: for n to 11 do P[n] := sort(expand(Q(n))) end do: for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 25 2011
STATUS
approved