A049210 a(n) = -Product_{k=0..n} (8*k-1); octo-factorial numbers.

1, 7, 105, 2415, 74865, 2919735, 137227545, 7547514975, 475493443425, 33760034483175, 2667042724170825, 232032717002861775, 22043108115271868625, 2270440135873002468375, 252018855081903273989625, 29990243754746489604765375, 3808760956852804179805202625

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..333

Index entries for sequences related to factorial numbers.

Formula

a(n) = 7*A034975(n) = (8*n-1)(!^8), n >= 1, a(0) = 1.

G.f.: 1/(1-7*x/(1-8*x/(1-15*x/(1-16*x/(1-23*x/(1-24*x/(1-31*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012

a(n) = (-1)^n*Sum_{k=0..n} 8^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 - 1/Q(0) )/x where Q(k) = 1 - x*(8*k-1)/(1 - x*(8*k+8)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013

a(n) = 8^n*Gamma(n+7/8)/Gamma(7/8). - R. J. Mathar, Mar 20 2013

E.g.f: (1-8*x)^(-7/8). - Vaclav Kotesovec, Jan 28 2015

G.f.: 1/(1-7*x-56*x^2/(1-23*x-240*x^2/(1-39*x-552*x^2/(1-55*x-992*x^2/(1-71*x-1560*x^2/(1-... )))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Dec 09 2020

G.f.: 1/G(0) where G(k) = 1 - (16*k+7)*x - 8*(k+1)*(8*k+7)*x^2/G(k+1); (continued fraction). - Nikolaos Pantelidis, Dec 19 2020

Sum_{n>=0} 1/a(n) = 1 + (e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). - Amiram Eldar, Dec 20 2022

Mathematica

FoldList[Times, 1, 8*Range[20]-1] (* Harvey P. Dale, Aug 03 2014 *)
CoefficientList[Series[(1-8*x)^(-7/8), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2015 *)

Prog

(PARI) a(n) = -prod(k=0, n, 8*k-1); \\ Michel Marcus, Jan 08 2015
(Sage) m=8; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # G. C. Greubel, Feb 16 2022
(Magma) m:=8; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..30]]; // G. C. Greubel, Feb 16 2022

Crossrefs

Cf. A034975, A048994, A051187.

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), this sequence (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

Sequence in context: A238464 A360347 A096131 * A167814 A209545 A002486

Adjacent sequences: A049207 A049208 A049209 * A049211 A049212 A049213

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved