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A167987
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Number of (undirected) cycles in the graph of the n-orthoplex, n>=2.
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5
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1, 63, 2766, 194650, 21086055, 3257119761, 679314442828, 183842034768036, 62630787876947325, 26224409462275175635, 13236607762537219815546, 7925653200467421739217118, 5554198822066977588903819331, 4503367772662184077396436475525, 4182811121982123218357983540881240
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OFFSET
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2,2
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COMMENTS
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The n-orthoplex, also known as the n-cross-polytope, is the dual of the n-cube.
A.k.a. number of (undirected) cycles in the n-cocktail party graph. - Eric W. Weisstein, Dec 29 2013
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LINKS
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FORMULA
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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
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EXAMPLE
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a(3) = 63, because in dimension n=3, the orthoplex is the octahedron, which has 63 cycles in its graph.
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MATHEMATICA
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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 *)
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PROG
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(PARI)
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
(Magma)
b:= func< n, k, j | (-1)^j*Binomial(n, j)*Binomial(2*(n-j), k-2*j)*2^(j-1)*Factorial(k-j-1) >;
A167986:= func< n, k | (&+[b(n, k, j): j in [0..Floor(k/2)]]) >;
(SageMath)
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)
@CachedFunction
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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