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

Author

Wolfdieter Lang

Keywords

• nonn

Links

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

Crossrefs

Cf. k-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542, A045754.

Cf. A048994, A051187, A113135.

Programs

GAP

List([0..20], n-> Product([0..n-1], j-> 8*j+1) ); # G. C. Greubel, Nov 11 2019

Magma

[1] cat [(&*[8*j+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019

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 *)

PARI

a(n)=prod(k=0, n, 8*k+1);

Sage

[product( (8*j+1) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 11 2019

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

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.