A062119 a(n) = n! * (n-1).

0, 2, 12, 72, 480, 3600, 30240, 282240, 2903040, 32659200, 399168000, 5269017600, 74724249600, 1133317785600, 18307441152000, 313841848320000, 5690998849536000, 108840352997376000, 2189611807358976000, 46225138155356160000, 1021818843434188800000

Offset: 1

Author

Olivier Gérard, Jun 13 2001

Keywords

• easy
• nonn

Comments

For n > 0, a(n) = number of permutations of length n+1 that have 2 predetermined elements nonadjacent; e.g., for n=2, the permutations with, say, 1 and 2 nonadjacent are 132 and 231, therefore a(2)=2. - Jon Perry, Jun 08 2003

Number of multiplications performed when computing the determinant of an n X n matrix by definition. - Mats Granvik, Sep 12 2008

Sum of the length of all cycles (excluding fixed points) in all permutations of [n]. - Olivier Gérard, Oct 23 2012

Number of permutations of n distinct objects (ABC...) 1 (one) times >>("-", A, AB, ABC, ABCD, ABCDE, ..., ABCDEFGHIJK, infinity) and one after the other to resemble motif: A (1) AB (1-1), AAB (2-1), AAAB (3-1), AAAAB (4-1), AAAAAB (5-1), AAAAAAB (6-1), AAAAAAAB (7-1), AAAAAAAAB (8-1) etc.,>> "1(one) fixed point". Example:motif: AAAB (or BBBA) 12 * one (1) fixed point etc. Let: AAAB ................ 'A'BCD 1. 'A'BDC 2. 'A'CBD 3. ACDB 'A'DBC 4. 'A'DCB B'A'CD 5. B'A'DC 6. BCAD 7. BCDA BD'A'C 8. BDCA C'A'BD 9. C'A'DB CB'A'D 10. CBDA CDAB CDBA D'A'BC 11. DACB DB'A'C 12. DBCA DCAB DCBA. - Zerinvary Lajos, Nov 27 2009 (does anybody understand what this is supposed to say? - Joerg Arndt, Jan 10 2015)

a(n) is the number of ways to arrange n books on two bookshelves so that each shelf receives at least one book. - Geoffrey Critzer, Feb 21 2010

a(n) = number whose factorial base representation (A007623) begins with digit {n-1} and is followed by n-1 zeros. Viewed in that base, this sequence looks like this: 0, 10, 200, 3000, 40000, 500000, 6000000, 70000000, 800000000, 9000000000, A0000000000, B00000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015

Links

• Harry J. Smith, Table of n, a(n) for n = 1..100
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Ugbene Ifeanyichukwu Jeff, Ogundele Olaniyi Suraju, and Ndubuisi Rich Ugochukwu, Digraph of the full transformation semigroup, J. Disc. Math. Sci. Crypt. (2021).
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• S. R. Schmitt Determinants [From _Mats Granvik_, Sep 12 2008]
• Index entries for sequences related to factorial base representation

Crossrefs

Column 2 of A257503 (apart from initial zero. Equally, row 2 of A257505).

Cf. A001286 (same sequence divided by 2).

Cf. A000142, A007623, A018931, A052849.

Cf. A001563. - Zerinvary Lajos, Aug 27 2008

Cf. sequences with formula (n + k)*n! listed in A282466.

Cf. A001113, A001620, A068985, A091725, A099285.

Programs

Haskell

a062119 n = (n - 1) * a000142 n  -- Reinhard Zumkeller, Aug 27 2012

Maple

G(x):=x^2/(1-x)^2: f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..19); # Zerinvary Lajos, Apr 01 2009

Mathematica

a[n_]:=1*(n+2)!-2*(n+1)!; (* or *) a[n_]:=n!*(n-1); (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)
Table[n!(n-1),{n,20}] (* Harvey P. Dale, Aug 29 2021 *)

PARI

{ f=1; for (n=1, 100, f*=n; write("b062119.txt", n, " ", f*(n - 1)) ) } \\ Harry J. Smith, Aug 02 2009

PARI

a(n) = n!*(n-1); \\ Altug Alkan, May 04 2018

Scheme

(define (A062119 n) (* (- n 1) (A000142 n))) ;; Antti Karttunen, May 07 2015

Formula

a(n) = n! * (n-1).

E.g.f.: x^2/(1-x)^2. - Geoffrey Critzer, Feb 21 2010

a(n) = 2 * A001286(n).

a(n) = A001563(n) - A000142(n). - Antti Karttunen, May 07 2015, hinted by crossref left by Lajos.

From Amiram Eldar, Jul 11 2020: (Start)

Sum_{n>=2} 1/a(n) = Ei(1) + 2 - e - gamma = A091725 + 2 - A001113 - A001620.

Sum_{n>=2} (-1)^n/a(n) = gamma - Ei(-1) - 1/e = A001620 + A099285 - A068985. (End)

Extensions

Last term a(19) corrected by Harry J. Smith, Aug 02 2009