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

%I A090932 #77 Nov 09 2024 06:31:23
%S A090932 1,1,1,3,6,30,90,630,2520,22680,113400,1247400,7484400,97297200,
%T A090932 681080400,10216206000,81729648000,1389404016000,12504636144000,
%U A090932 237588086736000,2375880867360000,49893498214560000,548828480360160000,12623055048283680000,151476660579404160000
%N A090932 a(n) = n! / 2^floor(n/2).
%C A090932 Number of permutations of the n-th row of Pascal's triangle.
%C A090932 Can be seen as the multiplicative equivalent to the generalized pentagonal numbers. - _Peter Luschny_, Oct 13 2012
%C A090932 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
%H A090932 Vincenzo Librandi, <a href="/A090932/b090932.txt">Table of n, a(n) for n = 0..200</a>
%H A090932 Rigoberto Flórez and Leandro Junes, <a href="http://www.emis.de/journals/INTEGERS/papers/l50/l50.Abstract.html">A relation between triangular numbers and prime numbers</a>, Integers, Vol. 12, No. 1 (2012), pp. 83-96.
%F A090932 a(n) = binomial(n-1, 2) * a(n-2).
%F A090932 E.g.f.: (1+x)/(1-1/2*x^2).
%F A090932 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
%F A090932 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
%F A090932 a(n) = (n+1)!/A093968(n+1). - _Anton Zakharov_, Jul 25 2016
%F A090932 a(n) ~ sqrt(2*Pi*n)*exp(-n)*n^n/2^floor(n/2). - _Ilya Gutkovskiy_, Jul 25 2016
%F A090932 From _Rigoberto Florez_, Apr 07 2017: (Start)
%F A090932 if n=2k, n! / 2^k = t(1)t(3)t(5)...t(2k-1),
%F A090932 if n=2k+1, n! / 2^k = t(2)t(4)t(6)...t(2k),
%F A090932 if n=2k, n! / 2^k = (t(k)-t(0))*(t(k)-t(1))*...*(t(k)-t(k-1)),
%F A090932 with t(i)= i-th triangular number. (End)
%F A090932 From _Amiram Eldar_, Feb 25 2022: (Start)
%F A090932 Sum_{n>=0} 1/a(n) = cosh(sqrt(2)) + sinh(sqrt(2))/sqrt(2).
%F A090932 Sum_{n>=0} (-1)^n/a(n) = cosh(sqrt(2)) - sinh(sqrt(2))/sqrt(2). (End)
%e A090932 From _Rigoberto Florez_, Apr 07 2017: (Start)
%e A090932 a(5) = 5!/2^2 = 120/4 = 30.
%e A090932 a(6) = 6!/2^3 = 1*6*15 = 90.
%e A090932 a(7) = 7!/2^3 = 3*10*21 = 630. (End)
%p A090932 a:= n-> n!/2^floor(n/2): seq(a(n), n=0..40);
%t A090932 Table[n!/2^Floor[n/2], {n, 0, 21}] (* _Michael De Vlieger_, Jul 25 2016 *)
%t A090932 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 *)
%o A090932 (PARI) a(n)=n!/2^floor(n/2)
%o A090932 (Magma) [Factorial(n) / 2^Floor(n/2): n in [0..25]]; // _Vincenzo Librandi_, May 14 2011
%o A090932 (Sage)
%o A090932 @CachedFunction
%o A090932 def A090932(n):
%o A090932     if n == 0 : return 1
%o A090932     fact = n//2 if is_even(n) else n
%o A090932     return fact * A090932(n-1)
%o A090932 [A090932(n) for n in (0..21)] # _Peter Luschny_, Oct 13 2012
%o A090932 (Python)
%o A090932 from math import factorial
%o A090932 def A090932(n): return factorial(n)>>(n>>1) # _Chai Wah Wu_, Jan 18 2023
%Y A090932 Cf. A052277, A007019.
%Y A090932 The function appears in several expansions: A009775, A046979, A046981, A007415, A007452.
%K A090932 nonn
%O A090932 0,4
%A A090932 _Jon Perry_, Feb 26 2004
%E A090932 Edited by _Ralf Stephan_, Sep 07 2004