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A129348 Number of (directed) Hamiltonian circuits in the cocktail party graph of order n.

%I A129348 #51 Feb 16 2025 08:33:05
%S A129348 0,2,32,1488,112512,12771840,2036229120,434469611520,119619533537280,
%T A129348 41303040523960320,17481826772405452800,8902337068174698086400,
%U A129348 5370014079716477003366400,3786918976243761421064601600,3087031512410698159166482022400,2880726660365605475506018320384000
%N A129348 Number of (directed) Hamiltonian circuits in the cocktail party graph of order n.
%C A129348 Also, the number of ways (up to rotations) to seat n married couples at a circular table with no spouses next to each other. Cf. A007060, A193639. - _Geoffrey Critzer_, Feb 09 2014
%C A129348 The cocktail party graph may also be called the n-octahedron, n-orthoplex or n-dimensional cross polytope. - _Andrew Howroyd_, May 14 2017
%H A129348 Max Alekseyev, <a href="/A129348/b129348.txt">Table of n, a(n) for n = 1..100</a>
%H A129348 Marko R. Riedel, Math.Stackexchange.com <a href="http://math.stackexchange.com/questions/1913728/">Proof of asymptotic (saddle point method) and closed form (inclusion-exclusion)</a>
%H A129348 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H A129348 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>
%F A129348 For n>=2, a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*(2*n-1-k)!*2^k. - _Geoffrey Critzer_, Feb 09 2014
%F A129348 Recurrence (for n>=4): (2*n-3)*a(n) = 2*(n-1)*(4*n^2 - 8*n + 5)*a(n-1) + 4*(n-2)*(n-1)*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Feb 09 2014
%F A129348 a(n) ~ sqrt(Pi) * 2^(2*n) * n^(2*n-1/2) / exp(2*n+1). - _Vaclav Kotesovec_, Feb 09 2014
%F A129348 For n>=2, a(n) = (-1 + 2 n)! Hypergeometric1F1[-n, 1 - 2 n, -2]. - _Eric W. Weisstein_, Mar 29 2014
%F A129348 a(n) = A003435(n) / (2*n) = A003436(n) * (n-1)! * 2^(n-1). - _Andrew Howroyd_, May 14 2017
%p A129348 a:= proc(n) option remember; `if`(n<3, n*(n-1),
%p A129348      ((136*n^3-608*n^2+762*n-470) *a(n-1)
%p A129348        +4*(n-2)*(14*n^2+29*n-193) *a(n-2)
%p A129348        -80*(n-2)*(n-3)*(n-4) *a(n-3)) /(34*n-101))
%p A129348     end:
%p A129348 seq(a(n), n=0..20);  # _Alois P. Heinz_, Feb 09 2014
%t A129348 Prepend[Table[Sum[(-1)^i Binomial[n, i] (2n - 1 - i)! 2^i, {i, 0, n}], {n, 2, 16}], 0] (* _Geoffrey Critzer_, Feb 09 2014 *)
%t A129348 Table[Piecewise[{{(-1 + 2 n)! Hypergeometric1F1[-n, 1 - 2 n, -2],
%t A129348     n > 1}}], {n, 16}] (* _Eric W. Weisstein_, Mar 29 2014 *)
%o A129348 (PARI) { A129348(n) = sum(m=0,n-1, sum(k=1,n-m, (-1)^k * binomial(n-1,m) * binomial(n-m-1,k-1) * 2^(k-1) * ([0,k-1,2*(n-m-k);1,k-2,2*(n-m-k);1,k-1,2*(n-m-k-1)]^(2*n))[1,1] ) + sum(k=0,n-m, (-1)^k * binomial(n-1,m) * binomial(n-m-1,k) * 2^(k-1) * ([0,k,2*(n-m-k-1);2,k-1,2*(n-m-k-1);2,k,2*(n-m-k-2)]^(2*n))[1,1] ) ) } \\ _Max Alekseyev_, Dec 22 2013
%Y A129348 Cf. A003435, A003436, A003437, A007060, A167987.
%K A129348 nonn
%O A129348 1,2
%A A129348 _Eric W. Weisstein_, Apr 10 2007
%E A129348 Terms a(6) onward from _Max Alekseyev_, Nov 10 2007