%I #36 Dec 21 2022 04:46:26

%S 1,4,52,1144,35464,1418560,69509440,4031547520,270113683840,

%T 20528639971840,1744934397606400,164023833375001600,

%U 16894454837625164800,1892178941814018457600,228953651959496233369600,29763974754734510338048000,4137192490908096936988672000

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

%H G. C. Greubel, <a href="/A144829/b144829.txt">Table of n, a(n) for n = 0..300</a>

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

%F 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

%F a(n) + (5-9*n)*a(n-1) = 0. - _R. J. Mathar_, Sep 04 2016

%F From _Vaclav Kotesovec_, Nov 29 2021: (Start)

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

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

%F From _G. C. Greubel_, May 26 2022: (Start)

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

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

%F 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

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

%t Table[4*9^(n-1)*Pochhammer[13/9, n-1], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 29 2021 *)

%o (PARI) a(n) = (-5)^n*sum(k=0, n, (9/5)^k*stirling(n+1,n+1-k, 1)); \\ _Michel Marcus_, Feb 20 2015

%o (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

%o (SageMath) [9^n*rising_factorial(4/9, n) for n in (0..30)] # _G. C. Greubel_, May 26 2022

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

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

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Sep 21 2008

%E a(9) originally given incorrectly as 20520639971840 corrected by _Peter Bala_, Feb 20 2015