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 A228308 Triangle read by rows: T(n,k) (n>=2, 1<=k<=n-1) is the number of unordered pairs of vertices at distances k in the odd graph O_n. 1
 3, 15, 30, 70, 210, 315, 315, 1260, 2520, 3780, 1386, 6930, 17325, 34650, 46200, 6006, 36036, 108108, 270270, 450450, 600600, 25740, 180180, 630630, 1891890, 3783780, 6306300, 7882875, 109395, 875160, 3500640, 12252240, 28588560, 57177120 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS Row n contains n-1 entries (n>=2). The odd graph O_n is a graph whose vertices represent the (n-1)-subsets of {1,2,...,2n-1} and two vertices are connected if and only if they correspond to disjoint subsets. It is a distance regular graph. The entries in row n are the coefficients of the Hosoya-Wiener polynomial of the odd graph O_n (n>=2). REFERENCES N. Biggs, Algebraic Graph Theory, Cambridge, 2nd. Ed., 1993, p. 161. R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer, The Wiener index of odd graphs, J. Indian. Inst. Sci., vol. 86, 2006, 527-531. LINKS Table of n, a(n) for n=2..35. R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer,, The Wiener index of odd graphs, J. Ind. Inst. Sci., vol. 86, no. 5, 2006. [Cached copy] Eric Weisstein's World of Mathematics, Odd Graph. FORMULA A formula is "hidden" in the Maple program. B(n) and C(n) are the intersection arrays of O_n while H(n) is the Hosoya-Wiener polynomial of O_n. EXAMPLE Row 2 has only one entry equal to 3; indeed, O_2 is the complete graph K_3, having 3 distances equal to 1. MAPLE B := proc (n) options operator, arrow: [seq(n-floor((1/2)*m), m = 1 .. n-1)] end proc: C := proc (n) options operator, arrow: [seq(ceil((1/2)*m), m = 1 .. n-1)] end proc: H := proc (n) options operator, arrow: (1/2)*binomial(2*n-1, n-1)*(sum((product(B(n)[r]/C(n)[r], r = 1 .. j))*t^j, j = 1 .. n-1)) end proc: for n from 2 to 10 do seq(coeff(H(n), t, k), k = 1 .. n-1) end do; # yields sequence in triangular form CROSSREFS Cf. A136328, A228309 Sequence in context: A290325 A271326 A106354 * A018972 A289962 A290069 Adjacent sequences: A228305 A228306 A228307 * A228309 A228310 A228311 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Aug 20 2013 STATUS approved

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Last modified October 3 10:37 EDT 2023. Contains 365861 sequences. (Running on oeis4.)