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%I #62 Dec 21 2023 21:09:20
%S 1,1,1,1,2,2,6,6,24,24,120,120,720,720,5040,5040,40320,40320,362880,
%T 362880,3628800,3628800,39916800,39916800,479001600,479001600,
%U 6227020800,6227020800,87178291200,87178291200,1307674368000,1307674368000,20922789888000,20922789888000
%N a(n) = floor(n/2)!.
%C This is the product of the first parts of the partitions (as nondecreasing list of parts) of n with exactly two positive integer parts, n > 1. - _Wesley Ivan Hurt_, Jan 25 2013
%H Vincenzo Librandi, <a href="/A081123/b081123.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = floor(n/2)!.
%F E.g.f.: 1+sqrt(Pi)/2*(x+2)*exp(x^2/4)*erf(x/2). - _Vladeta Jovovic_, Sep 25 2003
%F From _Sergei N. Gladkovskii_, Jul 28 2012: (Start)
%F G.f. G(0) where G(k) = 1 + x/(1 - x*(k+1)/( x*(k+1) + 1/G(k+1))); (continued fraction, 3rd kind, 3-step ).
%F E.g.f. 1 + sqrt(Pi)/2*(x+2)*exp(x^2/4)*erf(x/2) = 1 + x/(G(0)-x) where G(k) = 2*k + 1 + x - (2*k+1)*x/(x + 2 - 2*x/G(k+1)); (continued fraction, 1st kind, 2-step).
%F (End)
%F G.f.: U(0) where U(k) = 1 + x/(1 - x*(k+2)/(x*(k+2) + 1/U(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Oct 23 2012
%F G.f.: U(0) where U(k) = 1 + x/((2*k+1) - x*(2*k+1)/(x + 2*1/U(k+1))) ; (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Oct 23 2012
%F G.f.: 1 + x*G(0) where G(k) = 1 + x*(k+1)/(1 - x/(x + 1/G(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Nov 18 2012
%e a(8) = 24, since 8 has 4 nondecreasing partitions with exactly two positive integer parts: (1,7),(2,6),(3,5),(4,4). Multiplying the first parts of these partitions together, we get: (1)(2)(3)(4) = 4! = 24. - _Wesley Ivan Hurt_, Jun 03 2013
%p a:=n->floor(n/2)!: seq(a(k),k=1..70); # _Wesley Ivan Hurt_, Jun 03 2013
%t Table[(Floor[n/2])!, {n, 0, 40}] (* _Vincenzo Librandi_, Aug 06 2013 *)
%o (Magma) [Factorial(Floor(n/2)): n in [0..40]]; // _Vincenzo Librandi_, Aug 06 2013
%o (PARI) for(n=0,50, print1((n\2)!, ", ")) \\ _G. C. Greubel_, Aug 01 2017
%Y Cf. A000142, A004526, A081124, A081125.
%K easy,nonn
%O 0,5
%A _Paul Barry_, Mar 07 2003
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