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A045755
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8-fold factorials: a(n) = Product_{k=0..n-1} (8*k+1).
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29
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1, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625, 68586144251625, 5555477684381625, 494437513909964625, 47960438849266568625, 5035846079172989705625, 569050606946547836735625, 68855123440532288245010625, 8882310923828665183606370625
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n+1) = (8*n+1)(!^8).
a(n) = Sum_{k=0..n} (-8)^(n-k)*A048994(n, k); A048994 = Stirling-1 numbers.
E.g.f.: (1-8*x)^(-1/8).
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
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]
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
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
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
a(n) = 8^n * Gamma(n + 1/8) / Gamma(1/8). - Artur Jasinski,Aug 23 2016
a(n) ~ sqrt(2*Pi) * 8^n * n^(n - 3/8)/(Gamma(1/8)*exp(n)). - Ilya Gutkovskiy, Sep 10 2016
D-finite with recurrence: a(n) +(-8*n+7)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
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
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MAPLE
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a := n->product(8*k+1), k=0..(n-1));
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MATHEMATICA
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Table[8^n*Pochhammer[1/8, n], {n, 0, 20}] (* G. C. Greubel, Nov 11 2019 *)
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PROG
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(PARI) a(n)=prod(k=0, n, 8*k+1);
(Magma) [1] cat [(&*[8*j+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019
(Sage) [product( (8*j+1) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 11 2019
(GAP) List([0..20], n-> Product([0..n-1], j-> 8*j+1) ); # G. C. Greubel, Nov 11 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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