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 A005442 a(n) = n!*Fibonacci(n+1). (Formerly M3549) 18

%I M3549

%S 1,1,4,18,120,960,9360,105840,1370880,19958400,322963200,5748019200,

%T 111607372800,2347586841600,53178757632000,1290674601216000,

%U 33413695451136000,919096314200064000,26768324463648768000

%N a(n) = n!*Fibonacci(n+1).

%C 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 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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A005442/b005442.txt">Table of n, a(n) for n = 0..416</a>

%H P. R. J. Asveld & N. J. A. Sloane, <a href="/A005442/a005442.pdf">Correspondence, 1987</a>

%H P. R. J. Asveld, <a href="http://www.fq.math.ca/Scanned/25-4/asveld.pdf">A family of Fibonacci-like sequences</a>, Fib. Quart., 25 (1987), 81-83.

%H P. R. J. Asveld, <a href="http://www.fq.math.ca/Scanned/25-4/asveld.pdf">Another family of Fibonacci-like sequences</a>, Fib. Quart., 25 (1987), 361-364.

%H P. R. J. Asveld, <a href="http://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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=494">Encyclopedia of Combinatorial Structures 494</a>

%H 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 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F a(n) = A039948(n,0).

%F E.g.f.: 1/(1-x-x^2).

%F 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 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 Table[Fibonacci[n + 1]*n!, {n, 0, 20}] - _Zerinvary Lajos_, Jul 09 2009

%o (PARI) a(n) = n!*fibonacci(n+1) \\ _Charles R Greathouse IV_, Oct 03 2016

%Y Row sums of Fibonacci Jabotinsky-triangle A039692. A080599 and A052585.

%K nonn,easy

%O 0,3

%A _Simon Plouffe_