A001564 2nd differences of factorial numbers. (Formerly M2972 N1202)

1, 3, 14, 78, 504, 3720, 30960, 287280, 2943360, 33022080, 402796800, 5308934400, 75203251200, 1139544806400, 18394619443200, 315149522688000, 5711921639424000, 109196040425472000, 2196014181064704000, 46346783255764992000, 1024251745442365440000

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

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]

Index entries for sequences related to factorial numbers

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

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
(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: A366083 A368938 A242426 * A277132 A330074 A059276

Adjacent sequences: A001561 A001562 A001563 * A001565 A001566 A001567

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Comment edited by Franklin T. Adams-Watters, May 20 2010

Status

approved