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A001564
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2nd differences of factorial numbers.
(Formerly M2972 N1202)
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22
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1, 3, 14, 78, 504, 3720, 30960, 287280, 2943360, 33022080, 402796800, 5308934400, 75203251200, 1139544806400, 18394619443200, 315149522688000, 5711921639424000, 109196040425472000, 2196014181064704000, 46346783255764992000, 1024251745442365440000
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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 Janjic, Enumerative Formulas for Some Functions on Finite Sets
A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357. [From Emeric Deutsch, May 15 2010]
Index entries for sequences related to factorial numbers
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FORMULA
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a(n) = (n^2 + n + 1)*n! = A002061(n-1)*A000142(n). - Mitch Harris, Jul 10 2008
E.g.f.: (1+x^2)/(1-x)^3.
a(n) = A001563(n+1) - A001563(n). - Robert Israel, Apr 13 2015
a(n) = A306209(n+2,n). - Alois P. Heinz, Jan 29 2019
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MAPLE
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seq(factorial(n)*(n^2+n+1), n = 0 .. 20); # Emeric Deutsch, Apr 18 2009
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MATHEMATICA
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Range[0, 20]! CoefficientList[Series[(1+x^2)/(1-x)^3, {x, 0, 20}], x]
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PROG
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(PARI) Vec(serlaplace((1+x^2)/(1-x)^3 + O(x^30))) \\ Michel Marcus, Apr 10 2015
(MAGMA) [(n^2+n+1)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Apr 10 2015
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CROSSREFS
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Cf. A000142, A001563, A002061, A010027, A047920, A306209.
Sequence in context: A244507 A074538 A242426 * A277132 A330074 A059276
Adjacent sequences: A001561 A001562 A001563 * A001565 A001566 A001567
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Comment edited by Franklin T. Adams-Watters, May 20 2010
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STATUS
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approved
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