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A090932 a(n) = n! / 2^floor(n/2). 3
1, 1, 1, 3, 6, 30, 90, 630, 2520, 22680, 113400, 1247400, 7484400, 97297200, 681080400, 10216206000, 81729648000, 1389404016000, 12504636144000, 237588086736000, 2375880867360000, 49893498214560000 (list; graph; refs; listen; history; text; internal format)
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

R. Florez and L. Junes, A relation between triangular numbers and prime numbers, Integers 12 (2012), no. 1, 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)

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 *)

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

CROSSREFS

Cf. A052277, A007019.

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

Sequence in context: A125521 A211168 A215294 * A280981 A265376 A318431

Adjacent sequences:  A090929 A090930 A090931 * A090933 A090934 A090935

KEYWORD

nonn,changed

AUTHOR

Jon Perry, Feb 26 2004

EXTENSIONS

Edited by Ralf Stephan, Sep 07 2004

STATUS

approved

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Last modified December 6 22:42 EST 2021. Contains 349567 sequences. (Running on oeis4.)