OFFSET
0,4
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
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.
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)
EXAMPLE
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)
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 *)
PROG
(PARI) a(n)=n!/2^floor(n/2)
(Magma) [Factorial(n) / 2^Floor(n/2): n in [0..25]]; // Vincenzo Librandi, May 14 2011
(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
(Python)
from math import factorial
def A090932(n): return factorial(n)>>(n>>1) # Chai Wah Wu, Jan 18 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon Perry, Feb 26 2004
EXTENSIONS
Edited by Ralf Stephan, Sep 07 2004
STATUS
approved