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 A001564 2nd differences of factorial numbers. (Formerly M2972 N1202) 22
 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 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 , 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 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] FORMULA 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 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 (MAGMA) [(n^2+n+1)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Apr 10 2015 CROSSREFS Cf. A000142, A001563, A002061, A010027, A047920, A306209. Sequence in context: A244507 A074538 A242426 * A277132 A330074 A059276 Adjacent sequences:  A001561 A001562 A001563 * A001565 A001566 A001567 KEYWORD nonn,easy AUTHOR EXTENSIONS Comment edited by Franklin T. Adams-Watters, May 20 2010 STATUS approved

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Last modified April 13 00:24 EDT 2021. Contains 342934 sequences. (Running on oeis4.)