A045755 8-fold factorials: a(n) = Product_{k=0..n-1} (8*k+1).

1, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625, 68586144251625, 5555477684381625, 494437513909964625, 47960438849266568625, 5035846079172989705625, 569050606946547836735625, 68855123440532288245010625, 8882310923828665183606370625

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..330

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.

Cf. A048994, A051187, A113135.

Sequence in context: A217823 A361190 A113391 * A009037 A012148 A193540

Adjacent sequences: A045752 A045753 A045754 * A045756 A045757 A045758

Keyword

nonn

Author

Wolfdieter Lang

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