A084947 a(n) = Product_{i=0..n-1} (7*i+2).

1, 2, 18, 288, 6624, 198720, 7352640, 323516160, 16499324160, 956960801280, 62202452083200, 4478576549990400, 353807547449241600, 30427449080634777600, 2829752764499034316800, 282975276449903431680000, 30278354580139667189760000, 3451732422135922059632640000

Offset: 0

Author

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Keywords

• easy
• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..340

Crossrefs

Cf. A000079, A000142, A000165, A008544, A001813, A045754, A047055, A047657, A084942, A084947, A084948, A084949, A144739, A144827, A049209, A051188.

Programs

GAP

List([0..20], n-> Product([0..n-1], k-> 7*k+2) ); # G. C. Greubel, Aug 18 2019

Magma

[ 1 ] cat [ &*[ (7*k+2): k in [0..n-1] ]: n in [1..15] ]; // Klaus Brockhaus, Nov 10 2008

Maple

a := n->product(7*i+2,i=0..n-1); [seq(a(j),j=0..30)];

Mathematica

Join[{1},FoldList[Times,7*Range[0,15]+2]] (* Harvey P. Dale, Nov 27 2015 *)
Table[7^n*Pochhammer[2/7, n], {n,0,15}] (* G. C. Greubel, Aug 18 2019 *)

PARI

vector(20, n, n--; prod(k=0, n-1, 7*k+2)) \\ G. C. Greubel, Aug 18 2019

Sage

[product(7*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

Formula

a(n) = A084942(n)/A000142(n)*A000079(n) = 7^n*Pochhammer(2/7, n) = 7^n*Gamma(n+2/7)/Gamma(2/7).

D-finite with recurrence a(0) = 1; a(n) = (7*n - 5)*a(n-1) for n > 0. - Klaus Brockhaus, Nov 10 2008

G.f.: 1/(1-2*x/(1-7*x/(1-9*x/(1-14*x/(1-16*x/(1-21*x/(1-23*x/(1-28*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012

a(n) = (-5)^n*Sum_{k=0..n} (7/5)^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)^(2/7).

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

Sum_{n>=0} 1/a(n) = 1 + (e/7^5)^(1/7)*(Gamma(2/7) - Gamma(2/7, 1/7)). - Amiram Eldar, Dec 19 2022

Extensions

a(15) from Klaus Brockhaus, Nov 10 2008