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A005443 a(n) = n! * Fibonacci(n).

%I A005443 M2034 #54 Sep 01 2025 08:54:26
%S A005443 0,1,2,12,72,600,5760,65520,846720,12337920,199584000,3552595200,
%T A005443 68976230400,1450895846400,32866215782400,797681364480000,
%U A005443 20650793619456000,568032822669312000,16543733655601152000,508598164809326592000,16458582085314969600000
%N A005443 a(n) = n! * Fibonacci(n).
%C A005443 From _Enrique Navarrete_, Aug 28 2025: (Start)
%C A005443 Number of ways to seat n people on linearly ordered benches placing an odd number of people on each bench.
%C A005443 For example, a(7) = 65520 since the number of ways are (number of people in parentheses):
%C A005443  1 bench (7): 5040 ways;
%C A005443  3 benches (5,1,1): 15120 ways;
%C A005443  3 benches (3,3,1): 15120 ways;
%C A005443  5 benches (3,1,1,1,1): 25200 ways;
%C A005443  7 benches (1,1,1,1,1,1,1): 5040 ways.
%C A005443 If the benches are not linearly ordered the number of ways is A088009. (End)
%D A005443 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005443 Seiichi Manyama, <a href="/A005443/b005443.txt">Table of n, a(n) for n = 0..416</a>
%H A005443 P. R. J. Asveld & N. J. A. Sloane, <a href="/A005442/a005442.pdf">Correspondence, 1987</a>
%H A005443 P. R. J. Asveld, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/27-4/asveld.pdf">Fibonacci-like differential equations with a polynomial nonhomogeneous term</a>, Fib. Quart. 27 (1989), 303-309.
%H A005443 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=511">Encyclopedia of Combinatorial Structures 511</a>
%F A005443 a(n) = A039948(n, 1).
%F A005443 E.g.f.: x/(1-x-x^2). - _Geoffrey Critzer_, Sep 01 2013
%F A005443 a(n) = n*a(n-1) + n*(n-1)*a(n-2). - _G. C. Greubel_, Nov 20 2022
%p A005443 ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 1)}, labelled]: seq(combstruct[count](ZL, size=n), n=0..18); # _Zerinvary Lajos_, Mar 26 2008
%t A005443 Table[Fibonacci[n]*n!, {n, 0, 25}] (* _Zerinvary Lajos_, Jul 09 2009 *)
%o A005443 (PARI) a(n) = n!*fibonacci(n); \\ _Michel Marcus_, Oct 30 2015
%o A005443 (Magma) [Factorial(n)*Fibonacci(n): n in [0..30]]; // _G. C. Greubel_, Nov 20 2022
%o A005443 (SageMath) [fibonacci(n)*factorial(n) for n in range(31)] # _G. C. Greubel_, Nov 20 2022
%Y A005443 Cf. A000045, A000142, A005442, A039948, A088009.
%K A005443 nonn,easy,changed
%O A005443 0,3
%A A005443 _Simon Plouffe_
%E A005443 More terms from _James Sellers_, Dec 24 1999