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A005442 a(n) = n!*Fibonacci(n+1).

%I A005442 M3549 #73 Jan 05 2025 19:51:33
%S A005442 1,1,4,18,120,960,9360,105840,1370880,19958400,322963200,5748019200,
%T A005442 111607372800,2347586841600,53178757632000,1290674601216000,
%U A005442 33413695451136000,919096314200064000,26768324463648768000
%N A005442 a(n) = n!*Fibonacci(n+1).
%C A005442 Number of ways to use the elements of {1,...,n} once each to form a sequence of lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005
%C A005442 Number of pairs (p,S) where p is a permutation of {1,2,...,n} and S is a subset of {1,2,...,n} such that if s is in S then p(s) is not in S.  For example a(2) = 4 because we have (p=(1)(2), s={}); (p=(1,2), s={}); (p=(1,2), s={1}); (p=(1,2), s={2}) where p is given in cycle notation. - _Geoffrey Critzer_, Jul 01 2013
%C A005442 Another way to state the first comment: a(n) is the number of ways to partition [n] into blocks of size at most 2, order the blocks, and order the elements within each block. For example, a(5)=960 since there are 3 cases: 1) 1,2,3,4,5: 120 such ordered blocks, 1 way to order elements within blocks, hence 120 ways; 2) 12,3,4,5: 240 such ordered blocks, 2 ways to order elements within blocks, hence 480 ways; 3) 12,34,5: 90 such ordered blocks, 4 ways to order elements within blocks, hence 360 ways. - _Enrique Navarrete_, Sep 01 2023
%D A005442 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005442 Seiichi Manyama, <a href="/A005442/b005442.txt">Table of n, a(n) for n = 0..416</a>
%H A005442 P. R. J. Asveld & N. J. A. Sloane, <a href="/A005442/a005442.pdf">Correspondence, 1987</a>
%H A005442 P. R. J. Asveld, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-4/asveld.pdf">A family of Fibonacci-like sequences</a>, Fib. Quart., 25 (1987), 81-83.
%H A005442 P. R. J. Asveld, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-4/asveld.pdf">Another family of Fibonacci-like sequences</a>, Fib. Quart., 25 (1987), 361-364.
%H A005442 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 A005442 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=494">Encyclopedia of Combinatorial Structures 494</a>
%H A005442 Robert A. Proctor, <a href="http://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way For Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.
%H A005442 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A005442 a(n) = A039948(n,0).
%F A005442 E.g.f.: 1/(1-x-x^2).
%F A005442 D-finite with recurrence a(n) = n*a(n-1)+n*(n-1)*a(n-2). - Detlef Pauly (dettodet(AT)yahoo.de), Sep 22 2003
%F A005442 a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. Cf. A080599 and A052585. - _Peter Bala_, Dec 07 2011
%t A005442 Table[Fibonacci[n + 1]*n!, {n, 0, 20}] (* _Zerinvary Lajos_, Jul 09 2009 *)
%o A005442 (PARI) a(n) = n!*fibonacci(n+1) \\ _Charles R Greathouse IV_, Oct 03 2016
%o A005442 (Magma) [Factorial(n)*Fibonacci(n+1): n in [0..20]]; // _G. C. Greubel_, Nov 20 2022
%o A005442 (SageMath) [fibonacci(n+1)*factorial(n) for n in range(21)] # _G. C. Greubel_, Nov 20 2022
%Y A005442 Row sums of Fibonacci Jabotinsky-triangle A039692.
%Y A005442 Cf. A000045, A000142, A039948, A052585, A080599.
%K A005442 nonn,easy
%O A005442 0,3
%A A005442 _Simon Plouffe_
%E A005442 Comments from _Wolfdieter Lang_