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%I #62 Dec 22 2023 09:07:40
%S 0,2,12,72,480,3600,30240,282240,2903040,32659200,399168000,
%T 5269017600,74724249600,1133317785600,18307441152000,313841848320000,
%U 5690998849536000,108840352997376000,2189611807358976000,46225138155356160000,1021818843434188800000
%N a(n) = n! * (n-1).
%C 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
%C Number of multiplications performed when computing the determinant of an n X n matrix by definition. - _Mats Granvik_, Sep 12 2008
%C Sum of the length of all cycles (excluding fixed points) in all permutations of [n]. - _Olivier Gérard_, Oct 23 2012
%C 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)
%C 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
%C 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
%H Harry J. Smith, <a href="/A062119/b062119.txt">Table of n, a(n) for n = 1..100</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H Ugbene Ifeanyichukwu Jeff, Ogundele Olaniyi Suraju, and Ndubuisi Rich Ugochukwu, <a href="https://doi.org/10.1080/09720529.2020.1862955">Digraph of the full transformation semigroup</a>, J. Disc. Math. Sci. Crypt. (2021).
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H S. R. Schmitt <a href="https://web.archive.org/web/20100407230054/http://home.att.net/~srschmitt/script_determinant3.html">Determinants</a> [From _Mats Granvik_, Sep 12 2008]
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F a(n) = n! * (n-1).
%F E.g.f.: x^2/(1-x)^2. - _Geoffrey Critzer_, Feb 21 2010
%F a(n) = 2 * A001286(n).
%F a(n) = A001563(n) - A000142(n). - _Antti Karttunen_, May 07 2015, hinted by crossref left by Lajos.
%F From _Amiram Eldar_, Jul 11 2020: (Start)
%F Sum_{n>=2} 1/a(n) = Ei(1) + 2 - e - gamma = A091725 + 2 - A001113 - A001620.
%F Sum_{n>=2} (-1)^n/a(n) = gamma - Ei(-1) - 1/e = A001620 + A099285 - A068985. (End)
%p 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
%t a[n_]:=1*(n+2)!-2*(n+1)!; (* or *) a[n_]:=n!*(n-1); (* _Vladimir Joseph Stephan Orlovsky_, Dec 05 2008 *)
%t Table[n!(n-1),{n,20}] (* _Harvey P. Dale_, Aug 29 2021 *)
%o (PARI) { f=1; for (n=1, 100, f*=n; write("b062119.txt", n, " ", f*(n - 1)) ) } \\ _Harry J. Smith_, Aug 02 2009
%o (PARI) a(n) = n!*(n-1); \\ _Altug Alkan_, May 04 2018
%o (Haskell)
%o a062119 n = (n - 1) * a000142 n -- _Reinhard Zumkeller_, Aug 27 2012
%o (Scheme) (define (A062119 n) (* (- n 1) (A000142 n))) ;; _Antti Karttunen_, May 07 2015
%Y Column 2 of A257503 (apart from initial zero. Equally, row 2 of A257505).
%Y Cf. A001286 (same sequence divided by 2).
%Y Cf. A000142, A007623, A018931, A052849.
%Y Cf. A001563. - _Zerinvary Lajos_, Aug 27 2008
%Y Cf. sequences with formula (n + k)*n! listed in A282466.
%Y Cf. A001113, A001620, A068985, A091725, A099285.
%K easy,nonn
%O 1,2
%A _Olivier Gérard_, Jun 13 2001
%E Last term a(19) corrected by _Harry J. Smith_, Aug 02 2009
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