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A228308
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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.
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1
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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
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OFFSET
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2,1
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COMMENTS
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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).
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REFERENCES
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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.
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LINKS
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Eric Weisstein's World of Mathematics, Odd Graph.
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FORMULA
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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.
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EXAMPLE
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Row 2 has only one entry equal to 3; indeed, O_2 is the complete graph K_3, having 3 distances equal to 1.
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MAPLE
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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