A167986 Triangle T(n,k) = Number of k-cycles on the graph of an n-orthoplex. n>=2, k>=3.

0, 1, 8, 15, 24, 16, 32, 102, 288, 640, 960, 744, 80, 370, 1584, 5920, 18240, 43080, 69120, 56256, 160, 975, 5664, 30080, 141120, 564120, 1835520, 4542336, 7580160, 6385920, 280, 2121, 15624, 108080, 684480, 3876600, 19138560, 79805376

Offset

2,3

Comments

Row n contains 2n-2 elements.

The n-orthoplex is the dual polytope of the n-cube.

The orthoplex is also known as the cross-polytope.

Also the triangle of coefficients of the cocktail party graph cycle polynomials ordered from smallest to largest exponent starting with x^3. - Eric W. Weisstein

Links

Andrew Howroyd, Table of n, a(n) for n = 2..871

Eric Weisstein's World of Mathematics, Cross Polytope

Eric Weisstein's World of Mathematics, Cocktail Party Graph

Eric Weisstein's World of Mathematics, Cycle Polynomial

Formula

T(n,k) = 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

T(3,3) = 8, because in dimension n=3, the cross-polytope is the octahedron, which has 8 3-cycles in its graph.
Triangle starts
   0,   1;
   8,  15,   24,   16;
  32, 102,  288,  640,   960,   744;
  80, 370, 1584, 5920, 18240, 43080, 69120, 56256;
  ...
In terms of cycle polynomials:
   0*x^3 +    1*x^4;
   8*x^3 +  15*x^4 +  24*x^5 +  16*x^6;
  32*x^3 + 102*x^4 + 288*x^5 + 640*x^6 + 960*x^7 + 744*x^8;
  ...

Mathematica

T[n_, k_]:= Sum[(-1)^j*Binomial[n, j]*Binomial[2*(n-j), k-2*j]*2^j*(k - j - 1)!/2, {j, 0, Floor[k/2]}];
Table[T[n, k], {n, 2, 7}, {k, 3, 2*n}]//Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
Table[Binomial[2n, k]*Gamma[k]*HypergeometricPFQ[{(1-k)/2, -k/2}, {1 - k, 1/2 -n}, -2]/2, {n, 7}, {k, 3, 2n}]//Flatten (* Eric W. Weisstein, Mar 25 2020 *)

Prog

(PARI)
a(n, k)=sum(j=0, k\2, (-1)^j*binomial(n, j)*binomial(2*(n-j), k-2*j)*2^j*(k-j-1)!)/2;
for (n=2, 6, for (k=3, 2*n, print1(a(n, k), ", ")); print); \\ 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)]]) >;
[A167986(n, k): k in [3..2*n], n in [2..10]]; // 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)
flatten([[A167986(n, k) for k in range(3, 2*n+1)] for n in range(2, 11)]) # G. C. Greubel, Jan 17 2023

Crossrefs

Cf. A167987 (row sums).

Cf. A085452 (2k-cycles on graph of n-cube).

Cf. A144151 (k-cycles on (n-1)-simplex for k>3).

Sequence in context: A359000 A296542 A296546 * A015727 A034104 A034114

Adjacent sequences: A167983 A167984 A167985 * A167987 A167988 A167989

Keyword

nonn,tabf

Author

Andrew Weimholt, Nov 16 2009

Status

approved