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A033312
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a(n) = n! - 1.
(Formerly N1614)
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80
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0, 0, 1, 5, 23, 119, 719, 5039, 40319, 362879, 3628799, 39916799, 479001599, 6227020799, 87178291199, 1307674367999, 20922789887999, 355687428095999, 6402373705727999, 121645100408831999, 2432902008176639999
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OFFSET
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0,4
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COMMENTS
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a(n) gives the index number in any table of permutations of the entry in which the last n + 1 items are reversed. - Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004
a(n), n >= 1, has the factorial representation [n - 1, n - 2, ..., 1, 0]. The (unique) factorial representation of a number m from {0, 1, ... n! - 1} is m = sum(m_j(n)*j!, j = 0 .. n - 1) with m_j(n) from {0, 1, .., j}, n>=1. This is encoded as [m_{n-1},m_{n-2},...,m+1,m_0] with m_0=0. This can be interpreted as (D. N.) Lehmer code for the lexicographic rank of permutations of the symmetric group S_n (see the W. Lang link under A136663). The Lehmer code [n - 1, n - 2, ..., 1, 0] stands for the permutation [n, n - 1, ..., 1] (the last in lexicographic order). - Wolfdieter Lang, May 21 2008
For n >= 3: a(n) = numbers m for which there is one iteration {floor (r / k)} for k = n, n - 1, n - 2, ... 2 with property r mod k = k - 1 starting at r = m. For n = 5: a(5) = 119; floor (119 / 5) = 23, 119 mod 5 = 4; floor (23 / 4) = 5, 23 mod 4 = 3; floor (5 / 3) = 1, 5 mod 3 = 2; floor (1 / 2) = 0; 1 mod 2 = 1. - Jaroslav Krizek, Jan 23 2010
For n = 4, define the sum of all possible products of 1, 2, 3, 4 to be 1 + 2 + 3 + 4 add 1*2 + 1*3 + 1*4 add 2*3 + 2*4 + 3*4 add 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4 add 1*2*3*4. The sum of this is 119 = (4 + 1)! - 1. For n = 5 I get the sum 719 = (5 + 1)! - 1. The proof for the general case seems to follow by induction. - J. M. Bergot, Jan 10 2011
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 181.
Problem 598, J. Rec. Math., 11 (1978), 68-69.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..300
S. Legendre, P. Paclet, On the Permutations Generated by Cyclic Shift , J. Int. Seq. 14 (2011) # 11.3.2
G. P. Michon, Wilson's Theorem
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Andrew Walker, Factors of n! +- 1
Eric Weisstein's World of Mathematics, Factorial
Eric Weisstein's World of Mathematics, Permutation Pattern
Index entries for sequences related to factorial numbers
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FORMULA
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a(n) = Sum_{k = 1 .. n} (k-1)*(k-1)!.
a(n) = a(n - 1)*(n - 1) + a(n - 1) + n - 1, a(0) = 0. - Reinhard Zumkeller, Feb 03 2003
a(0) = a(1) = 0, a(n) = a(n - 1) * n + (n - 1) for n >= 2. - Jaroslav Krizek, Jan 23 2010
E.g.f.: 1/(1 - x) - exp(x). - Sergei N. Gladkovskii, Jun 29 2012
0 = 1 + a(n)*(+a(n+1) - a(n+2)) + a(n+1)*(+3 + a(n+1)) + a(n+2)*(-1) for n>=0. - Michael Somos, Feb 24 2017
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EXAMPLE
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G.f. = x^2 + 5*x^3 + 23*x^4 + 119*x^5 + 719*x^6 + 5039*x^7 + 40319*x^8 + ...
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MATHEMATICA
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FoldList[#1*#2 + #2 - 1 &, 0, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
Range[0, 19]! - 1 (* Alonso del Arte, Jan 24 2013 *)
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PROG
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(PARI) a(n)=n!-1 \\ Charles R Greathouse IV, Jul 19 2011
(MAGMA) [Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011
(Maxima) A033312(n):= n!-1$
makelist(A033312(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */
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CROSSREFS
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Cf. A000142, A001563 (first differences), A002582, A002982, A038507 (factorizations), A054415, A056110.
Sequence in context: A193704 A294356 A162815 * A151881 A229811 A121636
Adjacent sequences: A033309 A033310 A033311 * A033313 A033314 A033315
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane. This sequence appeared in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Eric W. Weisstein. Entry revised by N. J. A. Sloane, Jun 12 2012
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STATUS
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approved
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