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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.

%I A228308 #8 Feb 16 2025 08:33:20
%S A228308 3,15,30,70,210,315,315,1260,2520,3780,1386,6930,17325,34650,46200,
%T A228308 6006,36036,108108,270270,450450,600600,25740,180180,630630,1891890,
%U A228308 3783780,6306300,7882875,109395,875160,3500640,12252240,28588560,57177120
%N 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.
%C A228308 Row n contains n-1 entries (n>=2).
%C A228308 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.
%C A228308 The entries in row n are the coefficients of the Hosoya-Wiener polynomial of the odd graph O_n (n>=2).
%D A228308 N. Biggs, Algebraic Graph Theory, Cambridge, 2nd. Ed., 1993, p. 161.
%D A228308 R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer, The Wiener index of odd graphs, J. Indian. Inst. Sci., vol. 86, 2006, 527-531.
%H A228308 R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer,, <a href="/A136328/a136328.pdf">The Wiener index of odd graphs</a>, J. Ind. Inst. Sci., vol. 86, no. 5, 2006. [Cached copy]
%H A228308 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddGraph.html">Odd Graph</a>.
%F A228308 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.
%e A228308 Row 2 has only one entry equal to 3; indeed, O_2 is the complete graph K_3,  having 3 distances equal to 1.
%p A228308 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
%Y A228308 Cf. A136328, A228309
%K A228308 nonn,tabl
%O A228308 2,1
%A A228308 _Emeric Deutsch_, Aug 20 2013