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(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
[A167987(n) for n in range(2, 31)] # G. C. Greubel, Jan 17 2023
CROSSREFS
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