A049211 a(n) = Product_{k=1..n} (9*k - 1); 9-factorial numbers.

1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 101636305971200, 9045631231436800, 886471860680806400, 94852489092846284800, 11002888734770169036800, 1375361091846271129600000, 184298386307400331366400000, 26354669241958247385395200000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..325

Formula

a(n) = 8*A035022(n) = (9*n-1)(!^9), n >= 1, a(0) = 1.

a(n) = (-1)^n*Sum_{k=0..n} 9^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

a(n) = 9^n * Gamma(n+8/9) / Gamma(8/9). - Vaclav Kotesovec, Jan 28 2015

E.g.f: (1-9*x)^(-8/9). - Vaclav Kotesovec, Jan 28 2015

From Nikolaos Pantelidis, Dec 09 2020: (Start)

G.f.: 1/(1-8*x-72*x^2/(1-26*x-306*x^2/(1-44*x-702*x^2/(1-62*x-1260*x^2/(1-80*x-1980*x^2/(1-...)))))) (Jacobi continued fraction).

G.f.: 1/(1-8*x/(1-9*x/(1-17*x/(1-18*x/(1-26*x/(1-27*x/(1-35*x/(1-36*x/(1-44*x/(1-45*x/(1-...))))))))))) (Stieltjes continued fraction). (End)

From Nikolaos Pantelidis, Dec 19 2020: (Start)

G.f.: 1/G(0) where G(k) = 1 - (18*k+8)*x - 9*(k+1)*(9*k+8)*x^2/G(k+1) (continued fraction).

G.f.: 1/Q(0) where Q(k) = 1 - x*(9*k+8)/(1 - x*(9*k+9)/Q(k+1) ) (continued fraction). (End)

G.f.: hypergeometric2F0([1, 8/9], [--], 9*x). - G. C. Greubel, Feb 08 2022

Sum_{n>=0} 1/a(n) = 1 + (e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). - Amiram Eldar, Dec 21 2022

Mathematica

CoefficientList[Series[(1-9*x)^(-8/9), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2015 *)

Prog

(PARI) a(n) = prod(k=1, n, 9*k-1); \\ Michel Marcus, Jan 08 2015
(Magma) m:=9; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022
(Sage) m=9; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

Crossrefs

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), this sequence (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).

Cf. A035022, A048994.

Sequence in context: A132869 A036915 A238465 * A024283 A134053 A136472

Adjacent sequences: A049208 A049209 A049210 * A049212 A049213 A049214

Keyword

easy,nonn

Author

Wolfdieter Lang

Extensions

a(9) (originally given incorrectly as 1011636305971200) corrected by Peter Bala, Feb 20 2015

a(15)-a(16) from Vincenzo Librandi, Feb 20 2015

a(16) corrected and incorrect MAGMA program removed by Georg Fischer, May 10 2021

Status

approved