A144829 Partial products of successive terms of A017209; a(0)=1 .

1, 4, 52, 1144, 35464, 1418560, 69509440, 4031547520, 270113683840, 20528639971840, 1744934397606400, 164023833375001600, 16894454837625164800, 1892178941814018457600, 228953651959496233369600, 29763974754734510338048000, 4137192490908096936988672000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..300

Formula

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*9^(n-k).

a(n) = (-5)^n*Sum_{k=0..n} (9/5)^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

a(n) + (5-9*n)*a(n-1) = 0. - R. J. Mathar, Sep 04 2016

From Vaclav Kotesovec, Nov 29 2021: (Start)

a(n) = 9^n * Gamma(n + 4/9) / Gamma(4/9).

a(n) ~ sqrt(2*Pi) * 9^n * n^(n - 1/18) / (Gamma(4/9) * exp(n)). (End)

From G. C. Greubel, May 26 2022: (Start)

G.f.: hypergeometric2F0([1, 4/9], [], 9*x).

E.g.f.: (1-9*x)^(-4/9). (End)

Sum_{n>=0} 1/a(n) = 1 + (e/9^5)^(1/9)*(Gamma(4/9) - Gamma(4/9, 1/9)). - Amiram Eldar, Dec 21 2022

Example

a(0)=1, a(1)=4, a(2)=4*13=52, a(3)=4*13*22=1144, a(4)=4*13*22*31=35464, ...

Mathematica

Table[4*9^(n-1)*Pochhammer[13/9, n-1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 29 2021 *)

Prog

(PARI) a(n) = (-5)^n*sum(k=0, n, (9/5)^k*stirling(n+1, n+1-k, 1)); \\ Michel Marcus, Feb 20 2015
(Magma) [n le 2 select 4^(n-1) else (9*n-14)*Self(n-1): n in [1..30]]; // G. C. Greubel, May 26 2022
(SageMath) [9^n*rising_factorial(4/9, n) for n in (0..30)] # G. C. Greubel, May 26 2022

Crossrefs

Cf. A001715, A002866, A007559, A008546, A047053.

Cf. A048994, A049308, A132393, A144827, A144828.

Sequence in context: A091463 A354253 A354262 * A287699 A280571 A343429

Adjacent sequences: A144826 A144827 A144828 * A144830 A144831 A144832

Keyword

nonn,easy

Author

Philippe Deléham, Sep 21 2008

Extensions

a(9) originally given incorrectly as 20520639971840 corrected by Peter Bala, Feb 20 2015

Status

approved