1,1

The entries in row n are the coefficients of the Wiener polynomial of the benzenoid consisting of a linear chain of n hexagons.

Sum of entries in row n =(2n+1)(4n+1)=A014634(n).

Sum(k*T(n,k),k=1..2n+1)=the Wiener index of a benzenoid consisting of a linear chain of n hexagons=A143938(n).

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

For 1<=k<=2n+1, T(n,k) is given by: T(n,1)=5n+1, T(n,3)=9n-6, T(n,2p+1)=8n-8p+2, T(n,2p)=8n-8p+6.

G.f.=G(q,z)=qz/(6+6q-z+2qz+3q^2+q^2*z^2-q^4*z)/[(1-q^2*z)(1-z)^2].

T(1,2)=6 because in a hexagon there are 6 distances equal to 2.

Triangle starts:

6,6,3;

11,14,12,6,2;

16,22,21,14,10,6,2;

21,30,30,22,18,14,10,6,2;

T:=proc(n, k) if 2*n+1 < k then 0 elif k = 1 then 5*n+1 elif k = 3 then 9*n-6 elif `mod`(k, 2) = 0 then 8*n-4*k+6 else 8*n-4*k+6 end if end proc: for n to 8 do seq(T(n, k), k=1..2*n+1) end do; # yields sequence in triangular form

A014634, A143938

Sequence in context: A088454 A095228 A021605 this_sequence A019133 A094888 A159702

Adjacent sequences: A143934 A143935 A143936 this_sequence A143938 A143939 A143940

nonn,tabf

Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 06 2008