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A081125 a(n) = n! / floor(n/2)!.

%I A081125 #82 Jun 07 2025 08:29:50
%S A081125 1,1,2,6,12,60,120,840,1680,15120,30240,332640,665280,8648640,
%T A081125 17297280,259459200,518918400,8821612800,17643225600,335221286400,
%U A081125 670442572800,14079294028800,28158588057600,647647525324800,1295295050649600
%N A081125 a(n) = n! / floor(n/2)!.
%C A081125 Product of the largest parts in the partitions of n+1 into exactly two parts, n > 0. - _Wesley Ivan Hurt_, Jan 26 2013 (Clarified on Apr 20 2016)
%H A081125 Vincenzo Librandi, <a href="/A081125/b081125.txt">Table of n, a(n) for n = 0..400</a>
%H A081125 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A081125 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F A081125 E.g.f.: (1+x)*exp(x^2). - _Vladeta Jovovic_, Sep 24 2003
%F A081125 From _Peter Luschny_, Aug 07 2009: (Start)
%F A081125 a(n) = sqrt(n!*n$) where n$ denotes the swinging factorial (A056040).
%F A081125 a(n) = 2^n Gamma((n+1+(n mod 2))/2)/sqrt(Pi). (End)
%F A081125 E.g.f.: E(0) where E(k) = 1 + x/(1 - x/(x + (k+1)/E(k+1))) ; (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Sep 20 2012
%F A081125 G.f.: G(0) where G(k) =  1 + x*(2*k+1)/(1 - 2*x/(2*x + 1/G(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Nov 18 2012
%F A081125 D-finite with recurrence a(n) +2*a(n-1) -2*n*a(n-2) +4*(-n+2)*a(n-3) = 0. - _R. J. Mathar_, Nov 26 2012
%F A081125 From _Wesley Ivan Hurt_, Jun 06 2013: (Start)
%F A081125 a(n) = n!/(n-floor((n+1)/2))!.
%F A081125 a(n) = Product_{i = ceiling(n/2)..(n-1)} i. [Note: empty product = 1]
%F A081125 a(n) = P( n, floor((n+1)/2) ), where P(n,k) are the number of k-permutations of n objects. (End)
%F A081125 a(n) = n$*floor(n/2)! where n$ denotes the swinging factorial (A056040). - _Peter Luschny_, Oct 28 2013
%F A081125 From _Amiram Eldar_, Mar 10 2022: (Start)
%F A081125 Sum_{n>=0} 1/a(n) = 1 + (3/2)*exp(1/4)*sqrt(Pi)*erf(1/2).
%F A081125 Sum_{n>=0} (-1)^n/a(n) = 1 - (1/2)*exp(1/4)*sqrt(Pi)*erf(1/2). (End)
%e A081125 a(3) = 6 since 3+1 = 4 has two partitions into two parts, (3,1) and (2,2), and the product of the largest parts is 6. - _Wesley Ivan Hurt_, Jan 26 2013 (Clarified on Apr 20 2016)
%p A081125 Method 1)  a:=n->n!/floor(n/2)!; seq(a(k),k=0..40); # _Wesley Ivan Hurt_, Jun 03 2013
%p A081125 Method 2)  with(combinat, numbperm); seq(numbperm(k, floor((k+1)/2)), k = 0..40); # _Wesley Ivan Hurt_, Jun 06 2013
%t A081125 Table[n!/Floor[n/2]!, {n, 0, 30}] (* _Wesley Ivan Hurt_, Apr 20 2016 *)
%o A081125 (Magma) [Factorial(n)/(Factorial(Floor(n/2))): n in [0..30]]; // _Vincenzo Librandi_, Sep 13 2011
%o A081125 (PARI) a(n)=n!/(n\2)! \\ _Charles R Greathouse IV_, Sep 13 2011
%o A081125 (Sage)
%o A081125 def a(n): return rising_factorial(ceil(n/2),floor(n/2))
%o A081125 [a(n) for n in range(26)]  # _Peter Luschny_, Oct 09 2013
%o A081125 (Python)
%o A081125 from sympy import rf
%o A081125 def A081125(n): return rf((m:=n+1>>1)+(n+1&1),m) # _Chai Wah Wu_, Jul 22 2022
%Y A081125 Cf. A004526, A056040, A081123, A000407 (bisection), A001813 (bisection).
%K A081125 nonn,easy
%O A081125 0,3
%A A081125 _Paul Barry_, Mar 07 2003