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A147625 Octo-factorial numbers(4). 5

%I

%S 1,5,65,1365,39585,1464645,65909025,3493178325,213083877825,

%T 14702787569925,1132114642884225,96229744645159125,

%U 8949366251999798625,903885991451979661125,98523573068265783062625

%N Octo-factorial numbers(4).

%H G. C. Greubel, <a href="/A147625/b147625.txt">Table of n, a(n) for n = 1..330</a>

%F a(n+1) = Sum_{k=0..n} A132393(n,k)*5^k*8^(n-k). - _Philippe Deléham_, Nov 09 2008

%F a(n) = (-3)^n*Sum_{k=0..n} (8/3)^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 a(n) - (8*n-11)*a(n-1) = 0. - _R. J. Mathar_, Sep 04 2016

%F From _Benedict W. J. Irwin_, Sep 30 2016: (Start)

%F a(n) = Product_{k=0..n-1} (8*k+5).

%F a(n) = 8^n*Gamma(5/8 + n)/Gamma(5/8).

%F E.g.f: 1/(1 - 8*x)^(5/8). (End)

%F a(n)/n! ~ 8^n/(Gamma(5/8)*n^(3/8)). - _Vaclav Kotesovec_, Oct 04 2016

%p seq(8^(n-1)*pochhammer(5/8, n-1), n = 1..20); # _G. C. Greubel_, Dec 03 2019

%t Table[Product[(8k+5), {k, 0, n-1}], {n, 0, 20} (* _Benedict W. J. Irwin_, Sep 30 2016 *)

%o (PARI) vector(20, n, prod(j=0,n-2, 8*j+5) ) \\ _G. C. Greubel_, Dec 03 2019

%o (MAGMA) [Round(8^(n-1)*Gamma(n-1 +5/8)/Gamma(5/8)): n in [1..20]]; // _G. C. Greubel_, Dec 03 2019

%o (Sage) [8^(n-1)*rising_factorial(5/8, n-1) for n in (1..20)] # _G. C. Greubel_, Dec 03 2019

%Y Cf. A048994, A132393.

%K nonn,easy

%O 1,2

%A _Vladimir Joseph Stephan Orlovsky_, Nov 08 2008

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Last modified July 14 07:08 EDT 2020. Contains 335720 sequences. (Running on oeis4.)