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A074143
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a(1) = 1; a(n) = n * Sum_{k=1..n-1} a(k).
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13
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1, 2, 9, 48, 300, 2160, 17640, 161280, 1632960, 18144000, 219542400, 2874009600, 40475635200, 610248038400, 9807557760000, 167382319104000, 3023343138816000, 57621363351552000, 1155628453883904000, 24329020081766400000, 536454892802949120000
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OFFSET
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1,2
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COMMENTS
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a(n) is also the number of elements of the alternating semigroup (A^c_n) for F(n, p) if p = n - 1 (cf. A001710). - Bakare Gatta Naimat, Jan 15 2016
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..200
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Stephen Lipscomb, Symmetric inverse semigroups, Mathematical surveys and monographs, Vol.46 Amer. Math. Soc. (1996).
Michael Penn, Australian Mathematical Olympiad 2018 Question 5, Youtube video, 2020.
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FORMULA
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a(n) = n^2*a(n-1)/(n-1) for n > 2.
a(n) = n*ceiling[n!/2] = n*A001710(n-1) = A001710(n+1)-A001710(n) = ceiling[A001563(n)/2] - Henry Bottomley, Nov 27 2002
a(n) = ((n+1)!-n!)/2 for n > 1. - Vladimir Joseph Stephan Orlovsky, Apr 03 2011
G.f.: (U(0) + x)/(2*x) where U(k)= 1 - 1/(k+1 - x*(k+1)^2*(k+2)/(x*(k+1)*(k+2) - 1/U(k+1))) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Sep 27 2012
G.f.: 1/2 + Q(0), where Q(k)= 1 - 1/(k+2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3)-1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
a(n) = sum(j = 0..n, (-1)^(n-j)*binomial(n, j)*(j)^(n+1))/(n+1), n > 1, a(1) = 1. - Vladimir Kruchinin, Jun 01 2013
a(n) = numerator(n!/2*n). - Vincenzo Librandi, Apr 15 2014
a(n) is F(n;p) = n^2(n-1)!/2 if p = n-1 in A^c_n. For instance for n=4 and p=n-1: F(4; 4-1)= 4^2(4-1)!/2 = 16*6/2 = 48. - Bakare Gatta Naimat, Nov 18 2015
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MAPLE
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seq(sum(mul(j, j=3..n), k=1..n), n=1..19); # Zerinvary Lajos, Jun 01 2007
a := n -> `if`(n=1, 1, n!*n/2): seq(a(n), n=1..19); # Peter Luschny, Jan 22 2016
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MATHEMATICA
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A074143[1] = 1; A074143[n_] := A074143[n] = n * Sum[a[k], {k, n - 1}]; Array[A074143, 20] (* T. D. Noe, Apr 05 2011 *)
Table[Numerator[n!/2 n], {n, 40}] (* Vincenzo Librandi, Apr 15 2014)
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PROG
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(Magma) [Numerator(Factorial(n)/2*n): n in [1..30]]; // Vincenzo Librandi, Apr 15 2014
(SageMath)
def b(n): return 1/2 if (n==1) else n^2*b(n-1)/(n-1)
def A074143(n): return b(n) + int(n==1)/2
[A074143(n) for n in range(1, 41)] # G. C. Greubel, Nov 29 2022
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CROSSREFS
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A diagonal of A254040.
Cf. A001563, A001710.
Sequence in context: A171803 A100427 A214404 * A198892 A357790 A205571
Adjacent sequences: A074140 A074141 A074142 * A074144 A074145 A074146
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy, Aug 28 2002
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EXTENSIONS
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More terms from Henry Bottomley, Nov 27 2002
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STATUS
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approved
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