A192022 - OEIS

%I #9 Apr 09 2013 11:15:54

%S 1,0,3,2,1,5,5,4,1,7,8,8,4,1,9,11,12,8,4,1,11,14,16,12,8,4,1,13,17,20,

%T 16,12,8,4,1,15,20,24,20,16,12,8,4,1,17,23,28,24,20,16,12,8,4,1,19,26,

%U 32,28,24,20,16,12,8,4,1,21,29,36,32,28,24,20,16,12,8,4,1,23,32,40,36,32,28,24,20,16,12,8,4,1

%N Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the comb-shaped graph |_|_|...|_| with 2n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.

%C Row n>=2 contains n+1 entries.

%C Sum of entries in row n is n*(2n-1)=A000384(n) (the hexagonal numbers).

%C Sum(k*T(n,k),k>=1)=A192023(n) (the Wiener indices).

%F 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).

%e 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.

%e Triangle starts:

%e 1,0;

%e 3,2,1;

%e 5,5,4,1;

%e 7,8,8,4,1;

%e 9,11,12,8,4,1;

%p 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

%Y Cf. A000384, A192023

%K nonn,tabf

%O 1,3

%A _Emeric Deutsch_, Jun 24 2011