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 A180570 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the graph \|/_\/_\/_..._\/_\|/ having n nodes on the horizontal path. The entries in row n are the coefficients of the Wiener polynomial of the graph. 1
 7, 12, 9, 10, 18, 18, 9, 13, 24, 27, 18, 9, 16, 30, 36, 27, 18, 9, 19, 36, 45, 36, 27, 18, 9, 22, 42, 54, 45, 36, 27, 18, 9, 25, 48, 63, 54, 45, 36, 27, 18, 9, 28, 54, 72, 63, 54, 45, 36, 27, 18, 9, 31, 60, 81, 72, 63, 54, 45, 36, 27, 18, 9, 34, 66, 90, 81, 72, 63, 54, 45, 36, 27 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS Row n has n+1 entries. Sum of entries in row n = (2 + 9n + 9n^2)/2 =A060544(n+1). Sum_{k>=0} k*T(n,k) = A180571(n) (the Wiener indices of the graphs). REFERENCES I. Gutman, SL Lee, CH Chu. YLLuo, Indian J. Chem., 33A, 603. I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte fur Chemie, 131, 421-427 (see Eq. between (10) and (11); replace n with n+2). LINKS FORMULA The generating polynomial of row n is t*(9t^(n+2) - 3nt^3 - 8t^2 - 2t + 1 + 3n)/(1-t)^2. The bivariate g.f. is G = tz^2*(7 + 12t + 9t^2 - 4z - 13tz + 4tz^2 + 6t^2*z^2 - 12t^2*z)/((1-z)^2*(1-tz)). EXAMPLE T(2,3)=9 because in the graph \|/_\|/ there are 9 unordered pairs of vertices at distance 3. Triangle starts:    7, 12,  9;   10, 18, 18,  9;   13, 24, 27, 18,  9;   16, 30, 36, 27, 18,  9; MAPLE for n from 2 to 11 do P[n] := sort(expand(simplify(t*(9*t^(n+2)-3*n*t^3-8*t^2-2*t+1+3*n)/(1-t)^2))) end do: for n from 2 to 11 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form CROSSREFS Cf. A060544, A180571. Sequence in context: A300729 A152199 A293926 * A074474 A070420 A223423 Adjacent sequences:  A180567 A180568 A180569 * A180571 A180572 A180573 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Sep 16 2010 STATUS approved

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Last modified May 24 04:25 EDT 2019. Contains 323528 sequences. (Running on oeis4.)