OFFSET
0,3
COMMENTS
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..416
P. R. J. Asveld & N. J. A. Sloane, Correspondence, 1987
P. R. J. Asveld, A family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 81-83.
P. R. J. Asveld, Another family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 361-364.
P. R. J. Asveld, Fibonacci-like differential equations with a polynomial nonhomogeneous term, Fib. Quart. 27 (1989), 303-309.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 494
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
FORMULA
a(n) = A039948(n,0).
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
MATHEMATICA
Table[Fibonacci[n + 1]*n!, {n, 0, 20}] (* Zerinvary Lajos, Jul 09 2009 *)
PROG
(PARI) a(n) = n!*fibonacci(n+1) \\ Charles R Greathouse IV, Oct 03 2016
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Comments from Wolfdieter Lang
STATUS
approved