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

1, 63, 2766, 194650, 21086055, 3257119761, 679314442828, 183842034768036, 62630787876947325, 26224409462275175635, 13236607762537219815546, 7925653200467421739217118, 5554198822066977588903819331, 4503367772662184077396436475525, 4182811121982123218357983540881240

Offset: 2

Author

Andrew Weimholt, Nov 16 2009

Keywords

• nonn

Comments

Row sums of triangle in A167986.

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

Examples

			a(3) = 63, because in dimension n=3, the orthoplex is the octahedron, which has 63 cycles in its graph.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..100
• Eric Weisstein's World of Mathematics, Cocktail Party Graph
• Eric Weisstein's World of Mathematics, Cross Polytope
• Eric Weisstein's World of Mathematics, Graph Cycle

Crossrefs

Cf. A167986.

Programs

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)]]) >;
A167987:= func< n | (&+[A167986(n,k): k in [3..2*n]]) >;
[A167987(n): n in [2..30]]; // G. C. Greubel, Jan 17 2023

Mathematica

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 *)

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

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
def A167987(n): return sum(A167986(n,k) for k in range(3,2*n+1))
[A167987(n) for n in range(2,31)] # G. C. Greubel, Jan 17 2023

Formula

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

Extensions

a(8)-a(11) from Eric W. Weisstein, Dec 19 2013

a(12) from Eric W. Weisstein, Dec 21 2013

a(13) from Eric W. Weisstein, Jan 08 2014

a(14) from Eric W. Weisstein, Apr 09 2014

a(15)-a(16) from Andrew Howroyd, May 09 2017