A192022 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.

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, 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, 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

Offset: 1

Author

Emeric Deutsch, Jun 24 2011

Keywords

• nonn
• tabf

Comments

Row n>=2 contains n+1 entries.

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

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

Examples

			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;
		

Crossrefs

Cf. A000384, A192023

Programs

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

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).