A180573 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the sun graph on 2n nodes. The sun graph on 2n nodes is obtained by attaching n pendant edges to the cycle graph on n nodes.

6, 6, 3, 8, 10, 8, 2, 10, 15, 15, 5, 12, 18, 21, 12, 3, 14, 21, 28, 21, 7, 16, 24, 32, 28, 16, 4, 18, 27, 36, 36, 27, 9, 20, 30, 40, 40, 35, 20, 5, 22, 33, 44, 44, 44, 33, 11, 24, 36, 48, 48, 48, 42, 24, 6, 26, 39, 52, 52, 52, 52, 39, 13, 28, 42, 56, 56, 56, 56, 49, 28, 7, 30, 45, 60

Offset: 3

Author

Emeric Deutsch, Sep 19 2010

Keywords

• nonn
• tabf

Comments

Number of entries in row n = 2 + floor(n/2).

Sum of entries in row n = n(2n-1)=A000384(n).

Sum(k*T(n,k),k>=1) = A180574(n).

Examples

			Triangle starts:
6,6,3;
8,10,8,2;
10,15,15,5;
12,18,21,12,3;
		

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.
• D. Stevanovic, Hosoya polynomial of composite graphs, Discrete Math., 235 (2001), 237-244.

Crossrefs

Cf. A000384, A180574

Programs

Maple

P := proc (n) if `mod`(n, 2) = 0 then sort(expand(n*t*(1+t)^2*(sum(t^j, j = 0 .. (1/2)*n-1))+n*t-(1/2)*n*t^((1/2)*n)*(1+t)^2)) else sort(expand(n*t*(1+(1+t)^2*(sum(t^j, j = 0 .. (1/2)*n-3/2))))) end if end proc; for n from 3 to 15 do P(n) end do: for n from 3 to 15 do seq(coeff(P(n), t, i), i = 1 .. 2+floor((1/2)*n)) end do; # yields sequence in trianguklar form

Formula

The generating polynomial of row 2n is 2nt-nt^n*(1+t)^2+2nt(1+t)^2*sum(t^j, j=0..n-1); the generating polynomial of row 2n+1 is (2n+1)t[1+(1+t)^2*sum(t^j,j=0..n-1)]; these are the Wiener polynomials of the corresponding graphs.