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%I #60 Dec 20 2022 03:49:55
%S 1,1,9,153,3825,126225,5175225,253586025,14454403425,939536222625,
%T 68586144251625,5555477684381625,494437513909964625,
%U 47960438849266568625,5035846079172989705625,569050606946547836735625,68855123440532288245010625,8882310923828665183606370625
%N 8-fold factorials: a(n) = Product_{k=0..n-1} (8*k+1).
%H G. C. Greubel, <a href="/A045755/b045755.txt">Table of n, a(n) for n = 0..330</a>
%F a(n+1) = (8*n+1)(!^8).
%F a(n) = Sum_{k=0..n} (-8)^(n-k)*A048994(n, k); A048994 = Stirling-1 numbers.
%F E.g.f.: (1-8*x)^(-1/8).
%F G.f.: 1+x/(1-9x/(1-8x/(1-17x/(1-16x/(1-25x/(1-24x/(1-33x/(1-32x/(1-... (continued fraction). - _Philippe Deléham_, Jan 07 2012
%F a(n) = (-7)^n*Sum_{k=0..n} (8/7)^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.: 1/Q(0) where Q(k) = 1 - x*(8*k+1)/(1 - x*(8*k+8)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 20 2013
%F G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+1)/(2*x*(8*k+1) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013
%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(8*k+1)/(x*(8*k+1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 05 2013
%F a(n) = 8^n * Gamma(n + 1/8) / Gamma(1/8). - _Artur Jasinski_,Aug 23 2016
%F a(n) ~ sqrt(2*Pi) * 8^n * n^(n - 3/8)/(Gamma(1/8)*exp(n)). - _Ilya Gutkovskiy_, Sep 10 2016
%F D-finite with recurrence: a(n) +(-8*n+7)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020
%F Sum_{n>=0} 1/a(n) = 1 + (e/8^7)^(1/8)*(Gamma(1/8) - Gamma(1/8, 1/8)). - _Amiram Eldar_, Dec 20 2022
%p a := n->product(8*k+1), k=0..(n-1));
%t Table[8^n*Pochhammer[1/8, n], {n,0,20}] (* _G. C. Greubel_, Nov 11 2019 *)
%o (PARI) a(n)=prod(k=0, n, 8*k+1);
%o (Magma) [1] cat [(&*[8*j+1: j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Nov 11 2019
%o (Sage) [product( (8*j+1) for j in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Nov 11 2019
%o (GAP) List([0..20], n-> Product([0..n-1], j-> 8*j+1) ); # _G. C. Greubel_, Nov 11 2019
%Y Cf. k-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542, A045754.
%Y Cf. A048994, A051187, A113135.
%K nonn
%O 0,3
%A _Wolfdieter Lang_
%E Additional comments from _Philippe Deléham_ and _Paul D. Hanna_, Oct 29 2005
%E Edited by _N. J. A. Sloane_, Oct 14 2008 at the suggestion of _Artur Jasinski_.
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