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

%I A193624 #26 Sep 29 2023 13:09:45
%S A193624 1,0,72,37584,53529984,152458744320,766958183193600,
%T A193624 6236531290739312640,76788695692068062330880,
%U A193624 1361934174627779827740180480,33454092372947487842682293452800,1102556254139040688616563751190528000
%N A193624 Number of ways n triples can sit in a row without any siblings next to each other.
%H A193624 Andrew Woods, <a href="/A193624/b193624.txt">Table of n, a(n) for n = 0..101</a>
%F A193624 Lim_{n -> oo} a(n) -> (3n)!*exp(-2).
%F A193624 a(n) = A190826(n) * 6^n * n! for n >= 1. - _Nathaniel Johnston_, Aug 01 2011
%F A193624 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
%F A193624 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
%t A193624 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}];
%t A193624 Array[A193624, 30, 0] (* _G. C. Greubel_, Sep 22 2023 *)
%o A193624 (Magma)
%o A193624 B:=Binomial;
%o A193624 f:= func< n,j | (&+[B(n,k)*B(2*k,j)*(-3)^(n+k-j): k in [Ceiling(j/2)..n]]) >;
%o A193624 A193624:= func< n | (&+[Factorial(n+j)*f(n,j): j in [0..2*n]]) >;
%o A193624 [A193624(n): n in [0..30]]; // _G. C. Greubel_, Sep 22 2023
%o A193624 (SageMath)
%o A193624 b=binomial;
%o A193624 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))
%o A193624 def A193624(n): return sum(factorial(n+j)*f(j,n) for j in range(2*n+1))
%o A193624 [A193624(n) for n in range(31)] # _G. C. Greubel_, Sep 22 2023
%Y A193624 Cf. A007060, A190826, A330266.
%K A193624 nonn
%O A193624 0,3
%A A193624 _Andrew Woods_, Aug 01 2011