A090932 a(n) = n! / 2^floor(n/2).

1, 1, 1, 3, 6, 30, 90, 630, 2520, 22680, 113400, 1247400, 7484400, 97297200, 681080400, 10216206000, 81729648000, 1389404016000, 12504636144000, 237588086736000, 2375880867360000, 49893498214560000, 548828480360160000, 12623055048283680000, 151476660579404160000

Offset: 0

Author

Jon Perry, Feb 26 2004

Keywords

• nonn

Comments

Number of permutations of the n-th row of Pascal's triangle.

Can be seen as the multiplicative equivalent to the generalized pentagonal numbers. - Peter Luschny, Oct 13 2012

a(n) is the number of permutations of [n] in which all ascents start at an even position. For example, a(3) = 3 counts 213, 312, 321. - David Callan, Nov 25 2021

Examples

			From _Rigoberto Florez_, Apr 07 2017: (Start)
a(5) = 5!/2^2 = 120/4 = 30.
a(6) = 6!/2^3 = 1*6*15 = 90.
a(7) = 7!/2^3 = 3*10*21 = 630. (End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Rigoberto Flórez and Leandro Junes, A relation between triangular numbers and prime numbers, Integers, Vol. 12, No. 1 (2012), pp. 83-96.

Crossrefs

Cf. A052277, A007019.

The function appears in several expansions: A009775, A046979, A046981, A007415, A007452.

Programs

Magma

[Factorial(n) / 2^Floor(n/2): n in [0..25]]; // Vincenzo Librandi, May 14 2011

Maple

a:= n-> n!/2^floor(n/2): seq(a(n), n=0..40);

Mathematica

Table[n!/2^Floor[n/2], {n, 0, 21}] (* Michael De Vlieger, Jul 25 2016 *)
nxt[{n_,a_,b_}]:={n+1,b,a Binomial[n,2]}; NestList[nxt,{2,1,1},30][[All,2]] (* Harvey P. Dale, Aug 26 2022 *)

PARI

a(n)=n!/2^floor(n/2)

Python

from math import factorial
def A090932(n): return factorial(n)>>(n>>1) # Chai Wah Wu, Jan 18 2023

Sage

@CachedFunction
def A090932(n):
    if n == 0 : return 1
    fact = n//2 if is_even(n) else n
    return fact * A090932(n-1)
[A090932(n) for n in (0..21)] # Peter Luschny, Oct 13 2012

Formula

a(n) = binomial(n-1, 2) * a(n-2).

E.g.f.: (1+x)/(1-1/2*x^2).

E.g.f.: G(0) where G(k) = 1 + x/(1 - x/(x + 2/G(k+1) )) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 27 2012

G.f.: G(0), where G(k)= 1 + (2*k+1)*x/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 28 2013

a(n) = (n+1)!/A093968(n+1). - Anton Zakharov, Jul 25 2016

a(n) ~ sqrt(2*Pi*n)*exp(-n)*n^n/2^floor(n/2). - Ilya Gutkovskiy, Jul 25 2016

From Rigoberto Florez, Apr 07 2017: (Start)

if n=2k, n! / 2^k = t(1)t(3)t(5)...t(2k-1),

if n=2k+1, n! / 2^k = t(2)t(4)t(6)...t(2k),

if n=2k, n! / 2^k = (t(k)-t(0))*(t(k)-t(1))*...*(t(k)-t(k-1)),

with t(i)= i-th triangular number. (End)

From Amiram Eldar, Feb 25 2022: (Start)

Sum_{n>=0} 1/a(n) = cosh(sqrt(2)) + sinh(sqrt(2))/sqrt(2).

Sum_{n>=0} (-1)^n/a(n) = cosh(sqrt(2)) - sinh(sqrt(2))/sqrt(2). (End)

Extensions

Edited by Ralf Stephan, Sep 07 2004