A143939 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the cycle C_n (1 <= k <= floor(n/2)).

1, 3, 4, 2, 5, 5, 6, 6, 3, 7, 7, 7, 8, 8, 8, 4, 9, 9, 9, 9, 10, 10, 10, 10, 5, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 6, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 7, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 8, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18

Offset: 2

Author

Emeric Deutsch, Sep 06 2008

Keywords

• nonn
• tabf

Comments

Row n contains floor(n/2) entries.

The entries in row n are the coefficients of the Wiener polynomial of the cycle C_n.

Sum of entries in row n = n(n-1)/2 = A000217(n-1).

Sum_{k=1..floor(n/2)} k*T(n,k) = the Wiener index of the cycle C_n = A034828(n).

Examples

			T(4,2)=2 because in C_4 (a square) there are 2 distances equal to 2.
Triangle starts:
  1;
  3;
  4, 2;
  5, 5;
  6, 6, 3;
  7, 7, 7;
		

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.

Crossrefs

Cf. A000217, A034828.

Programs

Maple

P:=proc(n) if `mod`(n, 2)=0 then n*(sum(q^j,j=1..(1/2)*n-1))+(1/2)*n*q^((1/2)*n) else n*(sum(q^j,j=1..(1/2)*n-1/2)) end if end proc: for n from 2 to 18 do p[n]:=P(n) end do: for n from 2 to 18 do seq(coeff(p[n],q,j),j=1..floor((1/2)*n)) end do; # yields sequence in triangular form

Formula

T(2n+1,k) = 2n+1 (1<=k<=n); T(2n,k)=2n (1<=k<=n-1); T(2n,n)=n.

G.f. = G(q,z) = qz^2/(1+z-z^2-qz^3)/((1-qz^2)^2*(1-z)^2).