A180856 - OEIS

%I #8 Mar 29 2020 13:09:06

%S 9,6,12,16,15,20,10,18,24,24,21,28,28,14,24,32,32,32,27,36,36,36,18,

%T 30,40,40,40,40,33,44,44,44,44,22,36,48,48,48,48,48,39,52,52,52,52,52,

%U 26,42,56,56,56,56,56,56,45,60,60,60,60,60,60,30,48,64,64,64,64,64,64,64

%N Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the Moebius ladder M(n) (entries in row n are the coefficients of the corresponding Wiener polynomial). The Moebius ladder M(n) (n>=3) is the graph obtained from the cycle graph C(2n) by adding new edges joining each pair of opposite nodes.

%C Row n contains floor((n+1)/2) entries.

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

%C T(n,1)=3n = number of edges in the corresponding graph.

%C Sum(k*T(n,k), k>=1) = A180857(n).

%D N. Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MoebiusLadder.html">Moebius Ladder</a>.

%F The generating polynomial for row 2n+1 is n[3t+2t^{(n+1)/2}+4t^2*sum(t^j, j=0..(m-5)/2)] and for row 2n it is n[3t+4t^2*sum(t^j, j=0..(n-4)/2)] (these are also the Wiener polynomials of the corresponding Moebius ladders).

%e Triangle starts:

%e 9,6;

%e 12,16;

%e 15,20,10;

%e 18,24,24;

%e 21,28,28,14;

%e 24,32,32,32;

%p s := proc (m) options operator, arrow: sum(t^j, j = 0 .. m-2) end proc: P := proc (m) if `mod`(m, 2) = 0 then sort(expand(simplify(m*(3*t+4*t^2*s((1/2)*m))))) else sort(expand(simplify(m*(3*t+4*t^2*s((1/2)*m-1/2)+2*t^((1/2)*m+1/2))))) end if end proc: for m from 3 to 16 do P(m) end do: for n from 3 to 16 do seq(coeff(P(n), t, i), i = 1 .. floor((n+1)*1/2)) end do; # yields sequence in triangular form

%Y Cf. A000384, A180857.

%K nonn,tabf

%O 3,1

%A _Emeric Deutsch_, Sep 24 2010