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A045755 8-fold factorials: a(n) = Product_{k=0..n-1} (8*k+1). 29
1, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625, 68586144251625, 5555477684381625, 494437513909964625, 47960438849266568625, 5035846079172989705625, 569050606946547836735625, 68855123440532288245010625, 8882310923828665183606370625 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
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
MAPLE
a := n->product(8*k+1), k=0..(n-1));
MATHEMATICA
Table[8^n*Pochhammer[1/8, n], {n, 0, 20}] (* G. C. Greubel, Nov 11 2019 *)
PROG
(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
CROSSREFS
Cf. k-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542, A045754.
Sequence in context: A217823 A361190 A113391 * A009037 A012148 A193540
KEYWORD
nonn
AUTHOR
EXTENSIONS
Additional comments from Philippe Deléham and Paul D. Hanna, Oct 29 2005
Edited by N. J. A. Sloane, Oct 14 2008 at the suggestion of Artur Jasinski.
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)