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URL: https://oeis.org/A081125

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A081125
a(n) = n! / floor(n/2)!.
18
1, 1, 2, 6, 12, 60, 120, 840, 1680, 15120, 30240, 332640, 665280, 8648640, 17297280, 259459200, 518918400, 8821612800, 17643225600, 335221286400, 670442572800, 14079294028800, 28158588057600, 647647525324800, 1295295050649600
OFFSET
0,3
COMMENTS
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)
FORMULA
E.g.f.: (1+x)*exp(x^2). - Vladeta Jovovic, Sep 24 2003
From Peter Luschny, Aug 07 2009: (Start)
a(n) = sqrt(n!*n$) where n$ denotes the swinging factorial (A056040).
a(n) = 2^n Gamma((n+1+(n mod 2))/2)/sqrt(Pi). (End)
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
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
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
From Wesley Ivan Hurt, Jun 06 2013: (Start)
a(n) = n!/(n-floor((n+1)/2))!.
a(n) = Product_{i = ceiling(n/2)..(n-1)} i. [Note: empty product = 1]
a(n) = P( n, floor((n+1)/2) ), where P(n,k) are the number of k-permutations of n objects. (End)
a(n) = n$*floor(n/2)! where n$ denotes the swinging factorial (A056040). - Peter Luschny, Oct 28 2013
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + (3/2)*exp(1/4)*sqrt(Pi)*erf(1/2).
Sum_{n>=0} (-1)^n/a(n) = 1 - (1/2)*exp(1/4)*sqrt(Pi)*erf(1/2). (End)
EXAMPLE
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)
MAPLE
a:=n->n!/floor(n/2)!; seq(a(k), k=0..40); # Method 1. Wesley Ivan Hurt, Jun 03 2013
with(combinat, numbperm); seq(numbperm(k, floor((k+1)/2)), k = 0..40); # Method 2. Wesley Ivan Hurt, Jun 06 2013
MATHEMATICA
Table[n!/Floor[n/2]!, {n, 0, 30}] (* Wesley Ivan Hurt, Apr 20 2016 *)
PROG
(Magma) [Factorial(n)/(Factorial(Floor(n/2))): n in [0..30]]; // Vincenzo Librandi, Sep 13 2011
(PARI) a(n)=n!/(n\2)! \\ Charles R Greathouse IV, Sep 13 2011
(SageMath)
def a(n): return rising_factorial(ceil(n/2), floor(n/2))
[a(n) for n in range(26)] # Peter Luschny, Oct 09 2013
(Python)
from sympy import rf
def A081125(n): return rf((m:=n+1>>1)+(n+1&1), m) # Chai Wah Wu, Jul 22 2022
CROSSREFS
Cf. A004526, A056040, A081123, A000407 (bisection), A001813 (bisection).
Sequence in context: A191836 A072486 A096123 * A138570 A161887 A139315
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Mar 07 2003
STATUS
approved