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

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..338

Index entries for sequences related to factorial numbers.

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

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 *)

Prog

(PARI) a(n)=prod(k=0, n-1, 7*k+1)
(Magma) [1] cat [&*[7*j+1: j in [0..n-1]]: n in [1..20]]; // G. C. Greubel, Aug 21 2019
(Sage) [7^n*rising_factorial(1/7, n) for n in (0..20)] # G. C. Greubel, Aug 21 2019
(GAP) List([0..20], n-> Product([0..n-1], k-> 7*k+1) ); # G. C. Greubel, Aug 21 2019

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.

Sequence in context: A211825 A113383 A218671 * A339201 A360482 A229045

Adjacent sequences: A045751 A045752 A045753 * A045755 A045756 A045757

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 16 2008 at the suggestion of M. F. Hasler, Oct 14 2008

Corrected by Zerinvary Lajos, Apr 03 2009

Status

approved