A034912 One sixth of octo-factorial numbers.

1, 14, 308, 9240, 351120, 16151520, 872182080, 54075288960, 3785270227200, 295251077721600, 25391592684057600, 2386809712301414400, 243454590654744268800, 26780004972021869568000, 3160040586698580609024000, 398165113924021156737024000, 53354125265818835002761216000

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..333
• Index entries for sequences related to factorial numbers.

Crossrefs

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

Programs

Magma

[n le 1 select 1 else (8*n-2)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 20 2022

Maple

f:= proc(n) option remember; procname(n-1)*(8*n-2) end proc:
f(1):= 1:
map(f,[$1..20]); # Robert Israel, Mar 20 2018

Mathematica

Table[8^n*Pochhammer[3/4,n]/6, {n,40}] (* G. C. Greubel, Oct 20 2022 *)

SageMath

[8^n*rising_factorial(3/4,n)/6 for n in range(1,40)] # G. C. Greubel, Oct 20 2022

Formula

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).

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

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

From G. C. Greubel, Oct 20 2022: (Start)

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

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

From Amiram Eldar, Dec 20 2022: (Start)

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

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