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A005442
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a(n) = n!*Fibonacci(n+1).
(Formerly M3549)
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21
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1, 1, 4, 18, 120, 960, 9360, 105840, 1370880, 19958400, 322963200, 5748019200, 111607372800, 2347586841600, 53178757632000, 1290674601216000, 33413695451136000, 919096314200064000, 26768324463648768000
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OFFSET
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0,3
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COMMENTS
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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
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
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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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E.g.f.: 1/(1-x-x^2).
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
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
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MATHEMATICA
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PROG
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(Magma) [Factorial(n)*Fibonacci(n+1): n in [0..20]]; // G. C. Greubel, Nov 20 2022
(SageMath) [fibonacci(n+1)*factorial(n) for n in range(21)] # G. C. Greubel, Nov 20 2022
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CROSSREFS
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Row sums of Fibonacci Jabotinsky-triangle A039692.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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