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A167987 Number of (undirected) cycles in the graph of the n-orthoplex, n>=2.

%I A167987 #37 Feb 16 2025 08:33:11
%S A167987 1,63,2766,194650,21086055,3257119761,679314442828,183842034768036,
%T A167987 62630787876947325,26224409462275175635,13236607762537219815546,
%U A167987 7925653200467421739217118,5554198822066977588903819331,4503367772662184077396436475525,4182811121982123218357983540881240
%N A167987 Number of (undirected) cycles in the graph of the n-orthoplex, n>=2.
%C A167987 Row sums of triangle in A167986.
%C A167987 The n-orthoplex, also known as the n-cross-polytope, is the dual of the n-cube.
%C A167987 A.k.a. number of (undirected) cycles in the n-cocktail party graph. - _Eric W. Weisstein_, Dec 29 2013
%H A167987 Andrew Howroyd, <a href="/A167987/b167987.txt">Table of n, a(n) for n = 2..100</a>
%H A167987 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph </a>
%H A167987 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrossPolytope.html">Cross Polytope</a>
%H A167987 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%F A167987 a(n) = Sum_{k=3..2*n} Sum_{j=0..floor(k/2)} (-1)^j*binomial(n,j) * binomial(2*(n-j),k-2*j) * 2^j*(k-j-1)!/2. - _Andrew Howroyd_, May 09 2017
%e A167987 a(3) = 63, because in dimension n=3, the orthoplex is the octahedron, which has 63 cycles in its graph.
%t A167987 a[n_]:= Sum[Sum[(-1)^j*Binomial[n, j]*Binomial[2*(n-j), k-2*j]*2^j*(k - j-1)!, {j, 0, k/2}], {k, 3, 2 n}]/2; Array[a, 15, 2] (* _Jean-François Alcover_, Nov 01 2017, after _Andrew Howroyd_ *)
%o A167987 (PARI)
%o A167987 a(n)=sum(k=3,2*n, sum(j=0,k\2, (-1)^j*binomial(n,j)*binomial(2*(n-j),k-2*j)*2^j*(k-j-1)!))/2; \\ _Andrew Howroyd_, May 09 2017
%o A167987 (Magma)
%o A167987 b:= func< n,k,j | (-1)^j*Binomial(n,j)*Binomial(2*(n-j),k-2*j)*2^(j-1)*Factorial(k-j-1) >;
%o A167987 A167986:= func< n,k | (&+[b(n,k,j): j in [0..Floor(k/2)]]) >;
%o A167987 A167987:= func< n | (&+[A167986(n,k): k in [3..2*n]]) >;
%o A167987 [A167987(n): n in [2..30]]; // _G. C. Greubel_, Jan 17 2023
%o A167987 (SageMath)
%o A167987 def A167986(n,k): return simplify(binomial(2*n, k)*gamma(k)*hypergeometric([(1-k)/2, -k/2], [1-k, 1/2 -n], -2)/2)
%o A167987 @CachedFunction
%o A167987 def A167987(n): return sum(A167986(n,k) for k in range(3,2*n+1))
%o A167987 [A167987(n) for n in range(2,31)] # _G. C. Greubel_, Jan 17 2023
%Y A167987 Cf. A167986.
%K A167987 nonn
%O A167987 2,2
%A A167987 _Andrew Weimholt_, Nov 16 2009
%E A167987 a(8)-a(11) from _Eric W. Weisstein_, Dec 19 2013
%E A167987 a(12) from _Eric W. Weisstein_, Dec 21 2013
%E A167987 a(13) from _Eric W. Weisstein_, Jan 08 2014
%E A167987 a(14) from _Eric W. Weisstein_, Apr 09 2014
%E A167987 a(15)-a(16) from _Andrew Howroyd_, May 09 2017