A084949 a(n) = Product_{i=0..n-1} (9*i+2).

1, 2, 22, 440, 12760, 484880, 22789360, 1276204160, 82953270400, 6138542009600, 509498986796800, 46873906785305600, 4734264585315865600, 520769104384745216000, 61971523421784680704000, 7932354997988439130112000, 1086732634724416160825344000, 158662964669764759480500224000

Offset: 0

Author

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

Keywords

• easy
• nonn
• changed

Links

• Robert Israel, Table of n, a(n) for n = 0..329

Crossrefs

Cf. A000165, A008544, A001813, A047055, A047657, A084947, A084948.

Cf. A000079, A000142, A035012, A048994, A084944.

Programs

GAP

List([0..20], n-> Product([0..n-1], k-> 9*k+2) ); # G. C. Greubel, Aug 19 2019

Magma

[1] cat [(&*[9*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 19 2019

Maple

a:= n-> product(9*i+2,i=0..n-1); seq(a(j),j=0..20);

Mathematica

Table[9^n*Pochhammer[2/9, n], {n,0,20}] (* G. C. Greubel, Aug 19 2019 *)

PARI

vector(20, n, n--; prod(k=0, n-1, 9*k+2)) \\ G. C. Greubel, Aug 19 2019

Sage

[product(9*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 19 2019

Formula

a(n) = A084944(n)/A000142(n)*A000079(n) = 9^n*Pochhammer(2/9, n) = 9^n*Gamma(n+2/9)/Gamma(2/9).

a(n) = (-7)^n*Sum_{k=0..n} (9/7)^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

E.g.f.: (1-9*x)^(-2/9). - Robert Israel, Mar 22 2017

D-finite with recurrence: a(n) + (-9*n+7)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020

Sum_{n>=0} 1/a(n) = 1 + (e/9^7)^(1/9)*(Gamma(2/9) - Gamma(2/9, 1/9)). - Amiram Eldar, Dec 21 2022

a(n) ~ sqrt(2*Pi) * (9/e)^n * n^(n-5/18) / Gamma(2/9). - Amiram Eldar, Aug 30 2025