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
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