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A001564
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2nd differences of factorial numbers.
(Formerly M2972 N1202)
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16
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2009]
The average of the first n terms is n factorial. - Franklin T. Adams-Watters, May 20, 2010
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 12 2010: (Start)
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. (End)
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 (jmaxwellcampbell(AT)gmail.com), May 25, 2011]
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REFERENCES
| A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.
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).
A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 12 2010]
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LINKS
| Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for sequences related to factorial numbers
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FORMULA
| a(n) = (n^2 + n + 1)*n! = A002061(n-1)*A000142(n). - Mitch Harris (maharri(AT)gmail.com), Jul 10 2008
E.g.f.: (1+x^2)/(1-x)^3.
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MAPLE
| seq(factorial(n)*(n^2+n+1), n = 0 .. 20); [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2009]
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MATHEMATICA
| Range[0, 20]! CoefficientList[Series[(1+x^2)/(1-x)^3, {x, 0, 20}], x]
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CROSSREFS
| Cf. A047920.
Cf. A010027. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 12 2010]
Sequence in context: A048779 A052186 A074538 * A059276 A003169 A086621
Adjacent sequences: A001561 A001562 A001563 * A001565 A001566 A001567
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Comment edited by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 20 2010
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