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7-fold factorials: a(n) = Product_{k=0..n-1} (7*k+1).
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%I #70 Dec 19 2022 03:44:26

%S 1,1,8,120,2640,76560,2756160,118514880,5925744000,337767408000,

%T 21617114112000,1534815101952000,119715577952256000,

%U 10175824125941760000,936175819586641920000,92681406139077550080000,9824229050742220308480000,1110137882733870894858240000

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

%H G. C. Greubel, <a href="/A045754/b045754.txt">Table of n, a(n) for n = 0..338</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

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

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

%F 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

%F 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

%F 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

%F 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

%F a(n) = 7^n * Gamma(n + 1/7) / Gamma(1/7). - _Artur Jasinski_, Aug 23 2016

%F a(n) = A114799(7n-6). - _M. F. Hasler_, Feb 23 2018

%F D-finite with recurrence: a(n) +(-7*n+6)*a(n-1)=0. - _R. J. Mathar_, Jan 17 2020

%F 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

%p f := n->product( (7*k+1), k=0..(n-1));

%p 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

%t FoldList[Times, 1, 7Range[0, 20] + 1] (* _Harvey P. Dale_, Jan 21 2013 *)

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

%o (Magma) [1] cat [&*[7*j+1: j in [0..n-1]]: n in [1..20]]; // _G. C. Greubel_, Aug 21 2019

%o (Sage) [7^n*rising_factorial(1/7, n) for n in (0..20)] # _G. C. Greubel_, Aug 21 2019

%o (GAP) List([0..20], n-> Product([0..n-1], k-> 7*k+1) ); # _G. C. Greubel_, Aug 21 2019

%Y 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.

%Y See also A113134.

%Y Unsigned row sums of triangle A051186 (scaled Stirling1).

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

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

%K nonn

%O 0,3

%A _Wolfdieter Lang_

%E Additional comments from _Philippe Deléham_ and _Paul D. Hanna_, Oct 29 2005

%E Edited by _N. J. A. Sloane_, Oct 16 2008 at the suggestion of _M. F. Hasler_, Oct 14 2008

%E Corrected by _Zerinvary Lajos_, Apr 03 2009