A147626 - OEIS

%I #23 Dec 20 2022 03:52:40

%S 1,6,84,1848,55440,2106720,96909120,5233092480,324451733760,

%T 22711621363200,1771506466329600,152349556104345600,

%U 14320858273808486400,1460727543928465612800,160680029832131217408000,18960243520191483654144000,2388990683544126940422144000

%N Octo-factorial numbers (5).

%H G. C. Greubel, <a href="/A147626/b147626.txt">Table of n, a(n) for n = 1..325</a>

%F a(n+1) = Sum_{k=0..n} A132393(n,k)*6^k*8^(n-k). - _Philippe Deléham_, Nov 09 2008

%F a(n) = (-2)^n*Sum_{k=0..n} 4^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - _Mircea Merca_, May 03 2012

%F G.f.: 2*x/G(0), where G(k) = 1 + 1/(1 - 2*x*(8*k+6)/(2*x*(8*k+6) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013

%F From _G. C. Greubel_, Oct 21 2022: (Start)

%F a(n) = 8^n * Pochhammer(n, 3/4) = -2^(3*n-1) * Pochhammer(n, -1/4).

%F a(n) = (8*n - 10)*a(n-1). (End)

%F Sum_{n>=1} 1/a(n) = 1 + (e/8^2)^(1/8)*(Gamma(3/4) - Gamma(3/4, 1/8)). - _Amiram Eldar_, Dec 20 2022

%t s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,5,2*5!,8}];lst

%t Table[8^(n-1)*Pochhammer[3/4, n-1], {n,40}] (* _G. C. Greubel_, Oct 21 2022 *)

%o (Magma) [n le 1 select 1 else (8*n-10)*Self(n-1): n in [1..40]]; // _G. C. Greubel_, Oct 21 2022

%o (SageMath) [8^(n-1)*rising_factorial(3/4, n-1) for n in range(1,40)] # _G. C. Greubel_, Oct 21 2022

%Y Cf. A034908, A045755, A048994, A049210, A051189, A052714, A132393, A147625.

%K nonn

%O 1,2

%A _Vladimir Joseph Stephan Orlovsky_, Nov 08 2008