

A180856


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.


1



9, 6, 12, 16, 15, 20, 10, 18, 24, 24, 21, 28, 28, 14, 24, 32, 32, 32, 27, 36, 36, 36, 18, 30, 40, 40, 40, 40, 33, 44, 44, 44, 44, 22, 36, 48, 48, 48, 48, 48, 39, 52, 52, 52, 52, 52, 26, 42, 56, 56, 56, 56, 56, 56, 45, 60, 60, 60, 60, 60, 60, 30, 48, 64, 64, 64, 64, 64, 64, 64
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OFFSET

3,1


COMMENTS

Row n contains floor((n+1)/2) entries.
Sum of entries in row n is n(2n1)=A000384(n).
T(n,1)=3n = number of edges in the corresponding graph.
Sum(k*T(n,k), k>=1) = A180857(n).


REFERENCES

N. Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993.
B. E. Sagan, YN. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959969.


LINKS

Table of n, a(n) for n=3..72.
Eric Weisstein's World of Mathematics, Moebius Ladder.


FORMULA

The generating polynomial for row 2n+1 is n[3t+2t^{(n+1)/2}+4t^2*sum(t^j, j=0..(m5)/2)] and for row 2n it is n[3t+4t^2*sum(t^j, j=0..(n4)/2)] (these are also the Wiener polynomials of the corresponding Moebius ladders).


EXAMPLE

Triangle starts:
9,6;
12,16;
15,20,10;
18,24,24;
21,28,28,14;
24,32,32,32;


MAPLE

s := proc (m) options operator, arrow: sum(t^j, j = 0 .. m2) 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)*m1/2)+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


CROSSREFS

Cf. A000384, A180857
Sequence in context: A063602 A180572 A182873 * A166520 A102648 A272960
Adjacent sequences: A180853 A180854 A180855 * A180857 A180858 A180859


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Sep 24 2010


STATUS

approved



