cp's OEIS Frontend

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.

%I A180568 #7 Feb 16 2025 08:33:13
%S A180568 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,
%T A180568 32,46,44,35,26,17,8,2,37,54,53,44,35,26,17,8,2,42,62,62,53,44,35,26,
%U A180568 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
%N 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.
%C A180568 Row n contains n+1 entries.
%C A180568 Sum of entries in row n = (3/2)n(3n-1)=A062741(n).
%C A180568 The entries in row n are the coefficients of the Wiener polynomial of the grid P_3 x P_n.
%C A180568 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.
%C A180568 The average of all distances in the grid P_3 x P_n is (n+3)/3.
%H A180568 B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://users.math.msu.edu/users/sagan/Papers/Old/wpg-pub.pdf">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%H A180568 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.
%F A180568 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).
%F A180568 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.
%F A180568 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].
%e A180568 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.
%e A180568 Triangle starts:
%e A180568 2,1;
%e A180568 7,6,2;
%e A180568 12,14,8,2;
%e A180568 17,22,17,8,2;
%p A180568 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
%Y A180568 Cf. A180569
%K A180568 nonn,tabf
%O A180568 1,1
%A A180568 _Emeric Deutsch_, Sep 28 2010