|
|
A001564
|
|
2nd differences of factorial numbers.
(Formerly M2972 N1202)
|
|
26
|
|
|
1, 3, 14, 78, 504, 3720, 30960, 287280, 2943360, 33022080, 402796800, 5308934400, 75203251200, 1139544806400, 18394619443200, 315149522688000, 5711921639424000, 109196040425472000, 2196014181064704000, 46346783255764992000, 1024251745442365440000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
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
|
|
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
|
|
|
FORMULA
|
E.g.f.: (1+x^2)/(1-x)^3.
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]
|
|
PROG
|
(PARI) Vec(serlaplace((1+x^2)/(1-x)^3 + O(x^30))) \\ Michel Marcus, Apr 10 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|