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A193624
Number of ways n triples can sit in a row without any siblings next to each other.
6
1, 0, 72, 37584, 53529984, 152458744320, 766958183193600, 6236531290739312640, 76788695692068062330880, 1361934174627779827740180480, 33454092372947487842682293452800, 1102556254139040688616563751190528000
OFFSET
0,3
LINKS
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
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Woods, Aug 01 2011
STATUS
approved