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%I #13 Dec 13 2024 17:24:53
%S 10,72,448,2560,13824,71680,360448,1769472,8519680,40370176,188743680,
%T 872415232,3992977408,18119393280,81604378624,365072220160,
%U 1623497637888,7181185318912,31610959298560,138538465099776,604731395276800,2630031813640192
%N The hyper-Wiener index of the hypercube graph Q(n) (n>=2).
%C The hypercube graph Q(n) has as vertices the binary words of length n and an edge joins two vertices whenever the corresponding binary words differ in just one place.
%C Q(n) is distance-transitive and therefore also distance-regular. The intersection array is {n,n-1,n-2,...,1; 1,2,3,...,n-1,n}.
%D Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993 (p. 161).
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-48,64)
%F a(n) = 4^{n-2}*n*(3+n).
%F G.f.: 2*x^2*(5 - 24*x + 32*x^2)/(1-4*x)^3.
%F The Hosoya-Wiener polynomial of Q(n) is 2^{n-1}*((1+t)^n - 1).
%p a := proc (n) options operator, arrow: 4^(n-2)*n*(3+n) end proc: seq(a(n), n = 2 .. 25);
%t LinearRecurrence[{12,-48,64},{10,72,448},30] (* _Harvey P. Dale_, Dec 13 2024 *)
%Y Cf. A143376, A002697
%K nonn,easy
%O 2,1
%A _Emeric Deutsch_, Aug 20 2013