A180568 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the grid P_3 x P_n (1<=k<=n), where P_j denotes the path graph on j nodes.

2, 1, 7, 6, 2, 12, 14, 8, 2, 17, 22, 17, 8, 2, 22, 30, 26, 17, 8, 2, 27, 38, 35, 26, 17, 8, 2, 32, 46, 44, 35, 26, 17, 8, 2, 37, 54, 53, 44, 35, 26, 17, 8, 2, 42, 62, 62, 53, 44, 35, 26, 17, 8, 2, 47, 70, 71, 62, 53, 44, 35, 26, 17, 8, 2, 52, 78, 80, 71, 62, 53, 44, 35, 26, 17, 8, 2, 57

Offset: 1

Author

Emeric Deutsch, Sep 28 2010

Keywords

• nonn
• tabf

Comments

Row n contains n+1 entries.

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

The entries in row n are the coefficients of the Wiener polynomial of the grid P_3 x P_n.

Sum(k*T(n,k),k=1..n+1)=(1/2)n(n+3)(3n-1)=A180569(n) = the Wiener index of the grid P_3 x P_n.

The average of all distances in the grid P_3 x P_n is (n+3)/3.

Examples

			T(1,1)=2, T(1,2)=1 because in P_3 x P_1 = P_3 there are 2 pairs of nodes at distance 1 and one pair at distance 2.
Triangle starts:
2,1;
7,6,2;
12,14,8,2;
17,22,17,8,2;
		

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.
• Eric Weisstein's World of Mathematics, Grid Graph.

Crossrefs

Cf. A180569

Programs

Maple

p := proc (n) options operator, arrow: (t^(n+1)*(3+4*t+2*t^2)+(5*n-3)*t-(2*n+4)*t^2-(2*n+2)*t^3-n*t^4)/(1-t)^2 end proc: for n to 12 do f[n] := sort(expand(simplify(p(n)))) end do: for n to 12 do seq(coeff(f[n], t, k), k = 1 .. n+1) end do; # yields sequence in triangular form

Formula

The row generating polynomials p(n)=p(n,t) satisfy the recurrence relation p(n)=p(n-1)=2t+t^2+t(3+4t+2t^2)*sum(t^j,j=0..n-2) (these are the Wiener polynomials of the corresponding graphs).

The generating polynomial of row n is p(n; t)=[t^{n+1}*(3+4t+2t^2)+(5n-3)t-2(n+2)t^2-2(n+1)t^3-nt^4]/(1-t)^2.

G.f. = G(t,z)=Sum(T(n,k)*t^k*z^n, k>=1, n>=1) = tz(zt^2+2tz+t+3z+2)/[(1-tz)(1-z)^2].