cp's OEIS Frontend

A228309 The hyper-Wiener index of the odd graph O_n (n>=2).

%I A228309 #8 Feb 16 2025 08:33:20
%S A228309 3,105,2590,57015,1165626,22834812,433178460,8036703675,146451924190,
%T A228309 2632740298188,46790614294788,824017920352900,14397367664647800,
%U A228309 249906966022292400,4312825574857068600,74063143648813911075
%N A228309 The hyper-Wiener index of the odd graph O_n (n>=2).
%C A228309 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.
%D A228309 N. Biggs, Algebraic Graph Theory, Cambridge, 2nd. Ed., 1993, p. 161.
%D A228309 R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer, The Wiener index of odd graphs, J. Indian. Inst. Sci., vol. 86, 2006, 527-531.
%H A228309 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 A228309 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddGraph.html">Odd Graph</a>.
%F A228309 A formula is "hidden" in the Maple program. B(n) and C(n) are the intersection arrays of O_n, H(n) is the Hosoya-Wiener polynomial of O_n, and HWi(n) is the hyper-Wiener index of O_n.
%p A228309 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: HWi := proc (n) options operator, arrow: subs(t = 1, diff(H(n), t)+(1/2)*(diff(H(n), `$`(t, 2)))) end proc: seq(HWi(n), n = 2 .. 20);
%Y A228309 Cf. A136328, A228308
%K A228309 nonn
%O A228309 2,1
%A A228309 _Emeric Deutsch_, Aug 20 2013