A001564 2nd differences of factorial numbers.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

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

Number of permutations s of [n+2] where 2 designated elements are not mapped to themselves, e.g., s(1) != 1 and s(2) != 2. See Janjić article. - Benjamin Schreyer, May 07 2025

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 Janjić, Enumerative Formulae for Some Functions on Finite Sets
• A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000142, A001563, A002061, A010027, A047920, A306209.

Programs

Magma

[(n^2+n+1)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Apr 10 2015

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]
Differences[Range[0, 25]!, 2] (* Paolo Xausa, Jul 17 2025 *)

PARI

Vec(serlaplace((1+x^2)/(1-x)^3 + O(x^30))) \\ Michel Marcus, Apr 10 2015

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

Extensions

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