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A081125 a(n) = n! / floor(n/2)!. 5
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 (list; graph; refs; listen; history; text; internal format)
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)

REFERENCES

Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Index to divisibility sequences

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

Conjecture: 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

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

Method 1)  a:=n->n!/floor(n/2)!; seq(a(k), k=0..40); # Wesley Ivan Hurt, Jun 03 2013

Method 2)  with(combinat, numbperm); seq(numbperm(k, floor((k+1)/2)), k = 0..40); # 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

(Sage)

def a(n): return rising_factorial(ceil(n/2), floor(n/2))

[a(n) for n in range(26)]  # Peter Luschny, Oct 09 2013

CROSSREFS

Cf. A004526, A056040, A081123.

Sequence in context: A191836 A072486 A096123 * A138570 A161887 A139315

Adjacent sequences:  A081122 A081123 A081124 * A081126 A081127 A081128

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Mar 07 2003

STATUS

approved

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Last modified May 22 07:30 EDT 2019. Contains 323478 sequences. (Running on oeis4.)