A193624 Number of ways n triples can sit in a row without any siblings next to each other.

1, 0, 72, 37584, 53529984, 152458744320, 766958183193600, 6236531290739312640, 76788695692068062330880, 1361934174627779827740180480, 33454092372947487842682293452800, 1102556254139040688616563751190528000

Offset: 0

Author

Andrew Woods, Aug 01 2011

Keywords

• nonn

Links

• Andrew Woods, Table of n, a(n) for n = 0..101

Crossrefs

Cf. A007060, A190826, A330266.

Programs

Magma

B:=Binomial;
f:= func< n,j | (&+[B(n,k)*B(2*k,j)*(-3)^(n+k-j): k in [Ceiling(j/2)..n]]) >;
A193624:= func< n | (&+[Factorial(n+j)*f(n,j): j in [0..2*n]]) >;
[A193624(n): n in [0..30]]; // G. C. Greubel, Sep 22 2023

Mathematica

A193624[n_]:= Sum[(n+j)!*Binomial[n,k]*Binomial[2*k,j]*(-3)^(n+k-j), {j,0,2*n}, {k,Ceiling[j/2],n}];
Array[A193624, 30, 0] (* G. C. Greubel, Sep 22 2023 *)

SageMath

b=binomial;
def f(j,n): return sum(b(n,k)*b(2*k,j)*(-3)^(n+k-j) for k in range((j//2),n+1))
def A193624(n): return sum(factorial(n+j)*f(j,n) for j in range(2*n+1))
[A193624(n) for n in range(31)] # G. C. Greubel, Sep 22 2023

Formula

Lim_{n -> oo} a(n) -> (3n)!*exp(-2).

a(n) = A190826(n) * 6^n * n! for n >= 1. - Nathaniel Johnston, Aug 01 2011

a(n) -3*(9*n^3-9*n^2+8*n+8)*a(n-1) +108*(n-1)*(n^2-11*n+16)*a(n-2) +3024*(n-1)*(n-2)^2*a(n-3) -5184*(n-1)*(n-2)*(n-3)*a(n-4) = 0. - R. J. Mathar, May 23 2014

a(n) = Sum_{j=0..2*n} Sum_{k=ceiling(j/2)..n} (n+j)! * binomial(2*k, j) * binomial(n, k) * (-3)^(n+k-j). - G. C. Greubel, Sep 22 2023