

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)).


0



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
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OFFSET

2,2


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(n1)/2 = A000217(n1).
Sum_{k=1..floor(n/2)} k*T(n,k) = the Wiener index of the cycle C_n = A034828(n).


LINKS

Table of n, a(n) for n=2..79.
B. E. Sagan, YN. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959969.


FORMULA

T(2n+1,k) = 2n+1 (1<=k<=n); T(2n,k)=2n (1<=k<=n1); T(2n,n)=n.
G.f. = G(q,z) = qz^2/(1+zz^2qz^3)/((1qz^2)^2*(1z)^2).


EXAMPLE

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;


MAPLE

P:=proc(n) if `mod`(n, 2)=0 then n*(sum(q^j, j=1..(1/2)*n1))+(1/2)*n*q^((1/2)*n) else n*(sum(q^j, j=1..(1/2)*n1/2)) 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


CROSSREFS

Cf. A000217, A034828.
Sequence in context: A133570 A117041 A209688 * A197269 A201905 A138609
Adjacent sequences: A143936 A143937 A143938 * A143940 A143941 A143942


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Sep 06 2008


STATUS

approved



