OFFSET
1,3
COMMENTS
FORMULA
The generating polynomial of row n (i.e. the Wiener polynomial of the comb with 2n nodes) is n*t + t*(1+t)^2*(n*(1-t)-(1-t^n))/(1-t)^2 or, equivalently, n*t + t*(1+t)^2*Sum((n-j)*t^(j-1),j=1..n-1).
EXAMPLE
T(2,1)=3, T(2,2)=2, T(2,3)=1 because in the graph |_| there are 3 pairs of nodes at distance 1, 2 pairs at distance 2, and 1 pair at distance 3.
Triangle starts:
1,0;
3,2,1;
5,5,4,1;
7,8,8,4,1;
9,11,12,8,4,1;
MAPLE
Q := proc (n) options operator, arrow: n*t+t*(1+t)^2*(sum((n-j)*t^(j-1), j = 1 .. n-1)) end proc: for n to 12 do P[n] := sort(expand(Q(n))) end do: 1; for n from 2 to 12 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 24 2011
STATUS
approved