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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS Row n contains floor((n+1)/2) entries. Sum of entries in row n is n(2n-1)=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, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969. LINKS 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..(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). 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 .. 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 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

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