A045754 7-fold factorials: a(n) = Product_{k=0..n-1} (7*k+1).

1, 1, 8, 120, 2640, 76560, 2756160, 118514880, 5925744000, 337767408000, 21617114112000, 1534815101952000, 119715577952256000, 10175824125941760000, 936175819586641920000, 92681406139077550080000, 9824229050742220308480000, 1110137882733870894858240000

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..338
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A008542 (and A085158), A045755.

See also A113134.

Unsigned row sums of triangle A051186 (scaled Stirling1).

First column of triangle A132056 (S2(8)).

Cf. A114799, A084947, A144739, A144827, A147585, A049209, A051188.

Programs

GAP

List([0..20], n-> Product([0..n-1], k-> 7*k+1) ); # G. C. Greubel, Aug 21 2019

Magma

[1] cat [&*[7*j+1: j in [0..n-1]]: n in [1..20]]; // G. C. Greubel, Aug 21 2019

Maple

f := n->product( (7*k+1), k=0..(n-1));
G(x):=(1-7*x)^(-1/7): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..14); # Zerinvary Lajos, Apr 03 2009

Mathematica

FoldList[Times, 1, 7Range[0, 20] + 1] (* Harvey P. Dale, Jan 21 2013 *)

PARI

a(n)=prod(k=0,n-1,7*k+1)

Sage

[7^n*rising_factorial(1/7, n) for n in (0..20)] # G. C. Greubel, Aug 21 2019

Formula

a(n) = Sum_{k=0..n} (-7)^(n-k)*A048994(n, k), where A048994 = Stirling-1 numbers.

E.g.f.: (1-7*x)^(-1/7).

G.f.: 1/(1-x/(1-7*x/(1-8*x/(1-14*x/(1-15*x/(1-21*x/(1-22*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012

a(n) = (-6)^n*Sum_{k=0..n} (7/6)^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/G(0), where G(k)= 1 - x*(7*k+1)/(1 - x*(7*k+7)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(7*k+1)/(x*(7*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013

a(n) = 7^n * Gamma(n + 1/7) / Gamma(1/7). - Artur Jasinski, Aug 23 2016

a(n) = A114799(7n-6). - M. F. Hasler, Feb 23 2018

D-finite with recurrence: a(n) +(-7*n+6)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

Sum_{n>=0} 1/a(n) = 1 + (e/7^6)^(1/7)*(Gamma(1/7) - Gamma(1/7, 1/7)). - Amiram Eldar, Dec 19 2022

Extensions

Additional comments from Philippe Deléham and Paul D. Hanna, Oct 29 2005

Edited by N. J. A. Sloane, Oct 16 2008 at the suggestion of M. F. Hasler, Oct 14 2008

Corrected by Zerinvary Lajos, Apr 03 2009