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%I #72 Nov 29 2022 02:43:17
%S 1,2,9,48,300,2160,17640,161280,1632960,18144000,219542400,2874009600,
%T 40475635200,610248038400,9807557760000,167382319104000,
%U 3023343138816000,57621363351552000,1155628453883904000,24329020081766400000,536454892802949120000
%N a(1) = 1; a(n) = n * Sum_{k=1..n-1} a(k).
%C 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
%H Vincenzo Librandi, <a href="/A074143/b074143.txt">Table of n, a(n) for n = 1..200</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H Stephen Lipscomb, <a href="http://dx.doi.org/10.1090/surv/046">Symmetric inverse semigroups</a>, Mathematical surveys and monographs, Vol.46 Amer. Math. Soc. (1996).
%H Michael Penn, <a href="https://www.youtube.com/watch?v=zjEZyx4h1GQ">Australian Mathematical Olympiad 2018 Question 5</a>, Youtube video, 2020.
%F a(n) = n^2*a(n-1)/(n-1) for n > 2.
%F a(n) = n*ceiling[n!/2] = n*A001710(n-1) = A001710(n+1)-A001710(n) = ceiling[A001563(n)/2] - _Henry Bottomley_, Nov 27 2002
%F a(n) = ((n+1)!-n!)/2 for n > 1. - _Vladimir Joseph Stephan Orlovsky_, Apr 03 2011
%F 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
%F 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
%F 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
%F a(n) = numerator(n!/2*n). - _Vincenzo Librandi_, Apr 15 2014
%F 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
%p seq(sum(mul(j,j=3..n), k=1..n), n=1..19); # _Zerinvary Lajos_, Jun 01 2007
%p a := n -> `if`(n=1,1,n!*n/2): seq(a(n), n=1..19); # _Peter Luschny_, Jan 22 2016
%t A074143[1] = 1; A074143[n_] := A074143[n] = n * Sum[a[k], {k, n - 1}]; Array[A074143, 20] (* _T. D. Noe_, Apr 05 2011 *)
%t Table[Numerator[n!/2 n], {n, 40}] (* _Vincenzo Librandi_, Apr 15 2014)
%o (Magma) [Numerator(Factorial(n)/2*n): n in [1..30]]; // _Vincenzo Librandi_, Apr 15 2014
%o (SageMath)
%o def b(n): return 1/2 if (n==1) else n^2*b(n-1)/(n-1)
%o def A074143(n): return b(n) + int(n==1)/2
%o [A074143(n) for n in range(1,41)] # _G. C. Greubel_, Nov 29 2022
%Y A diagonal of A254040.
%Y Cf. A001563, A001710.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Aug 28 2002
%E More terms from _Henry Bottomley_, Nov 27 2002
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