A034912 - OEIS

%I #38 Dec 20 2022 08:02:03

%S 1,14,308,9240,351120,16151520,872182080,54075288960,3785270227200,

%T 295251077721600,25391592684057600,2386809712301414400,

%U 243454590654744268800,26780004972021869568000,3160040586698580609024000,398165113924021156737024000,53354125265818835002761216000

%N One sixth of octo-factorial numbers.

%H Robert Israel, <a href="/A034912/b034912.txt">Table of n, a(n) for n = 1..333</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%F 6*a(n) = (8*n-2)(!^8) = Product_{j=1..n} (8*j - 2) = 2^n*3*A034176(n), where 3*A034176(n) = (4*n-1)(!^4) = Product_{j=1..n} (4*j - 1).

%F E.g.f.: (-1+(1-8*x)^(-3/4))/6.

%F G.f.: x/(1-14*x/(1-8*x/(1-22*x/(1-16*x/(1-30*x/(1-24*x/(1-38*x/(1-32*x/(1-...(continued fraction). - _Philippe Deléham_, Jan 07 2012

%F From _G. C. Greubel_, Oct 20 2022: (Start)

%F a(n) = (1/6) * 8^n * Pochhammer(n, 3/4).

%F a(n) = 2*(4*n - 1)*a(n-1). (End)

%F From _Amiram Eldar_, Dec 20 2022: (Start)

%F a(n) = A147626(n+1)/6.

%F Sum_{n>=1} 1/a(n) = 6*(e/8^2)^(1/8)*(Gamma(3/4) - Gamma(3/4, 1/8)). (End)

%p f:= proc(n) option remember; procname(n-1)*(8*n-2) end proc:

%p f(1):= 1:

%p map(f,[$1..20]); # _Robert Israel_, Mar 20 2018

%t Table[8^n*Pochhammer[3/4,n]/6, {n,40}] (* _G. C. Greubel_, Oct 20 2022 *)

%o (Magma) [n le 1 select 1 else (8*n-2)*Self(n-1): n in [1..40]]; // _G. C. Greubel_, Oct 20 2022

%o (SageMath) [8^n*rising_factorial(3/4,n)/6 for n in range(1,40)] # _G. C. Greubel_, Oct 20 2022

%Y Cf. A034176, A034908, A034909, A034910, A034911, A045755, A147626.

%K easy,nonn

%O 1,2

%A _Wolfdieter Lang_