A144739 - OEIS

%I #41 Dec 19 2022 03:43:28

%S 1,3,30,510,12240,379440,14418720,648842400,33739804800,1990648483200,

%T 131382799891200,9590944392057600,767275551364608000,

%U 66752972968720896000,6274779459059764224000,633752725365036186624000,68445294339423908155392000,7871208849033749437870080000

%N 7-factorial numbers A114799(7*n+3): Partial products of A017017(k) = 7*k+3, a(0) = 1.

%H G. C. Greubel, <a href="/A144739/b144739.txt">Table of n, a(n) for n = 0..335</a>

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

%F a(n) = Sum_{k=0..n} A132393(n,k)*3^k*7^(n-k).

%F G.f.: 1/(1-3*x/(1-7*x/(1-10*x/(1-14*x/(1-17*x/(1-21*x/(1-24*x/(1-... (continued fraction). - _Philippe Deléham_, Jan 08 2012

%F a(n) = (-4)^n*Sum_{k=0..n} (7/4)^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 From _Ilya Gutkovskiy_, Mar 23 2017: (Start)

%F E.g.f.: 1/(1 - 7*x)^(3/7).

%F a(n) ~ sqrt(2*Pi)*7^n*n^n/(exp(n)*n^(1/14)*Gamma(3/7)). (End)

%F a(n) = A114799(7*n-4). - _M. F. Hasler_, Feb 23 2018

%F D-finite with recurrence: a(n) +(-7*n+4)*a(n-1)=0. - _R. J. Mathar_, Feb 21 2020

%F Sum_{n>=0} 1/a(n) = 1 + (e/7^4)^(1/7)*(Gamma(3/7) - Gamma(3/7, 1/7)). - _Amiram Eldar_, Dec 19 2022

%e a(0)=1, a(1)=3, a(2)=3*10=30, a(3)=3*10*17=510, a(4)=3*10*17*24=12240, ...

%p a:= n-> product(7*j+3, j=0..n-1); seq(a(n), n=0..20); # _G. C. Greubel_, Aug 19 2019

%t Table[7^n*Pochhammer[3/7, n], {n,0,20}] (* _G. C. Greubel_, Aug 19 2019 *)

%o (Magma) [ 1 ] cat [ &*[ (7*k+3): k in [0..n] ]: n in [0..20] ]; // _Klaus Brockhaus_, Nov 10 2008

%o (PARI) a(n)=prod(i=1,n,7*i-4) \\ _Charles R Greathouse IV_, Jul 02 2013

%o (Sage) [product(7*k+3 for k in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Aug 19 2019

%o (GAP) List([0..20], n-> Product([0..n-1], k-> 7*k+3) ); # _G. C. Greubel_, Aug 19 2019

%Y Cf. A114799, A001710, A001147, A032031, A008545, A047056, A011781, A045754, A084947, A144827, A147585, A049209, A051188.

%K nonn

%O 0,2

%A _Philippe Deléham_, Sep 20 2008