A144739 7-factorial numbers A114799(7*n+3): Partial products of A017017(k) = 7*k+3, a(0) = 1.

1, 3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800, 1990648483200, 131382799891200, 9590944392057600, 767275551364608000, 66752972968720896000, 6274779459059764224000, 633752725365036186624000, 68445294339423908155392000, 7871208849033749437870080000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..335

Index entries for sequences related to factorial numbers.

Formula

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

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

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

From Ilya Gutkovskiy, Mar 23 2017: (Start)

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

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

a(n) = A114799(7*n-4). - M. F. Hasler, Feb 23 2018

D-finite with recurrence: a(n) +(-7*n+4)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

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

Example

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

Maple

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

Mathematica

Table[7^n*Pochhammer[3/7, n], {n, 0, 20}] (* G. C. Greubel, Aug 19 2019 *)

Prog

(Magma) [ 1 ] cat [ &*[ (7*k+3): k in [0..n] ]: n in [0..20] ]; // Klaus Brockhaus, Nov 10 2008
(PARI) a(n)=prod(i=1, n, 7*i-4) \\ Charles R Greathouse IV, Jul 02 2013
(Sage) [product(7*k+3 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 19 2019
(GAP) List([0..20], n-> Product([0..n-1], k-> 7*k+3) ); # G. C. Greubel, Aug 19 2019

Crossrefs

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

Sequence in context: A064352 A366002 A338278 * A185827 A366009 A370910

Adjacent sequences: A144736 A144737 A144738 * A144740 A144741 A144742

Keyword

nonn

Author

Philippe Deléham, Sep 20 2008

Status

approved