A192024 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the double-comb graph \/\/\/...\/_\/ with 3n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.

2, 1, 5, 6, 4, 8, 12, 12, 4, 11, 18, 21, 12, 4, 14, 24, 30, 21, 12, 4, 17, 30, 39, 30, 21, 12, 4, 20, 36, 48, 39, 30, 21, 12, 4, 23, 42, 57, 48, 39, 30, 21, 12, 4, 26, 48, 66, 57, 48, 39, 30, 21, 12, 4, 29, 54, 75, 66, 57, 48, 39, 30, 21, 12, 4, 32, 60, 84, 75, 66, 57, 48, 39, 30, 21, 12, 4

Offset: 1

Author

Emeric Deutsch, Jun 25 2011

Keywords

• nonn
• tabf

Comments

Row n contains n+1 entries.

Sum of entries in row n is 3*n*(3*n-1)/2=A062741(n).

Sum(k*T(n,k),k>=1)=A192025(n) (the Wiener indices).

Examples

			T(2,1)=5, T(2,2)=6, T(2,3)=4 because in the graph \/_\/ there are 5 pairs of nodes at distance 1, 6 pairs at distance 2, and 4 pairs at distance 3.
Triangle starts:
2,1;
5,6,4;
8,12,12,4;
11,18,21,12,4;
		

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. A062741, A192025

Programs

Maple

Q := proc (n) options operator, arrow: n*(t^2+2*t)+t*(1+2*t)^2*(sum((n-j)*t^(j-1), j = 1 .. n-1)) end proc: for n to 11 do P[n] := sort(expand(Q(n))) end do: for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form

Formula

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

G.f. of column 1: z*(2+z)/(1-z)^2.

G.f. of column 2: z*(1+4*z+z^2)/(1-z)^2.

G.f. of column k>=3: z^(k-1)*(4+4*z+z^2)/(1-z)^2.

The generating polynomial of row n (i.e. the Wiener polynomial of the double-comb with 3n nodes) is n*(2*t +t^2)+t*(1+2*t)^2*(n*(1-t)-(1-t^n))/(1-t)^2 or, equivalently, n*(2*t+t^2)+t*(1+2*t)^2*Sum((n-j)*t^(j-1), j=1..n-1).