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

2,2

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

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A167986.

Sequence in context: A094938 A006110 A132051 * A273437 A069381 A051589

Adjacent sequences: A167984 A167985 A167986 * A167988 A167989 A167990

Keyword

nonn

Author

Andrew Weimholt, Nov 16 2009

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

Status

approved