OFFSET
0,2
COMMENTS
a(n) is also the number of isolated fixed points (i.e. adjacent fixed points are not isolated) in all permutations of [n+2]. Example: a(2)=14 because we have (the isolated fixed points are marked) 1'423, 1'324', 1'342, 1'43'2, 413'2, 3124', 42'13, 2314', 243'1, 32'14', 32'41. - _Emeric Deutsch_, Apr 18 2009
The average of the first n terms is n factorial. - _Franklin T. Adams-Watters_, May 20 2010
Number of blocks in all permutations of [n+1]. A block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67. Example: a(2)=14 because the permutations of [3], separated into blocks, are 123, 1-3-2, 2-1-3, 23-1, 3-12, 3-2-1 with 1+3+3+2+2+3=14 blocks. - _Emeric Deutsch_, Jul 12 2010
a(n) equals n+1 times the permanent of the (n+1) X (n+1) matrix with 1/(n+1) in the top right corner and 1's everywhere else. - _John M. Campbell_, May 25 2011
Number of permutations s of [n+2] where 2 designated elements are not mapped to themselves, e.g., s(1) != 1 and s(2) != 2. See Janjić article. - _Benjamin Schreyer_, May 07 2025
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.
Milan Janjić, Enumerative Formulae for Some Functions on Finite Sets
A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357.
FORMULA
E.g.f.: (1+x^2)/(1-x)^3.
a(n) = A306209(n+2,n). - _Alois P. Heinz_, Jan 29 2019
D-finite with recurrence a(n) +(-n-3)*a(n-1) +(n-1)*a(n-2)=0. - _R. J. Mathar_, Jul 01 2022
MAPLE
seq(factorial(n)*(n^2+n+1), n = 0 .. 20); # _Emeric Deutsch_, Apr 18 2009
MATHEMATICA
Range[0, 20]! CoefficientList[Series[(1+x^2)/(1-x)^3, {x, 0, 20}], x]
(* Alternative: *)
Differences[Range[0, 25]!, 2] (* _Paolo Xausa_, Jul 17 2025 *)
PROG
(PARI) Vec(serlaplace((1+x^2)/(1-x)^3 + O(x^30))) \\ _Michel Marcus_, Apr 10 2015
(PARI) apply( {A001564(n)=(1+n+n^2)*n!}, [0..22]) \\ _M. F. Hasler_, Jun 01 2026
(Magma) [(n^2+n+1)*Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Apr 10 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Comment edited by Franklin T. Adams-Watters, May 20 2010
STATUS
approved
