%I #20 Dec 21 2022 04:46:22

%S 1,5,70,1610,51520,2112320,105616000,6231344000,423731392000,

%T 32627317184000,2805949277824000,266565181393280000,

%U 27722778864901120000,3132674011733826560000,382186229431526840320000,50066396055530016081920000,7009295447774202251468800000

%N 9-factorial numbers (4).

%H G. C. Greubel, <a href="/A147629/b147629.txt">Table of n, a(n) for n = 1..300</a>

%F a(n+1) = Sum_{k=0..n} A132393(n,k)*5^k*9^(n-k). - _Philippe Deléham_, Nov 09 2008

%F From _R. J. Mathar_, Nov 09 2008: (Start)

%F a(n) = a(n-1) + (4 + 9*(n-2))*a(n-1) = (9*n-13)*a(n-1).

%F a(n) = 9^(n-1)*Gamma(n-4/9)/Gamma(5/9).

%F G.f.: z*2F0(5/9,1; -; 9*z). (End)

%F a(n) = (-4)^n*Sum_{k=0..n} (9/4)^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 Sum_{n>=1} 1/a(n) = 1 + (e/9^4)^(1/9)*(Gamma(5/9) - Gamma(5/9, 1/9)). - _Amiram Eldar_, Dec 21 2022

%p seq(9^(n-1)*pochhammer(5/9, n-1), n = 1..20); # _G. C. Greubel_, Dec 03 2019

%t Table[9^(n-1)*Pochhammer[5/9, n-1], {n,20}] (* _G. C. Greubel_, Dec 03 2019 *)

%o (PARI) vector(20, n, prod(j=0,n-2, 9*j+5) ) \\ _G. C. Greubel_, Dec 03 2019

%o (Magma) [Round(9^(n-1)*Gamma(n-1 +5/9)/Gamma(5/9)): n in [1..20]]; // _G. C. Greubel_, Dec 03 2019

%o (Sage) [9^(n-1)*rising_factorial(5/9, n-1) for n in (1..20)] # _G. C. Greubel_, Dec 03 2019

%Y Cf. A035012, A035013, A035017, A035018, A035020, A035022, A035023, A045756, A048994, A049211, A051232, A053116, A132393.

%K nonn

%O 1,2

%A _Vladimir Joseph Stephan Orlovsky_, Nov 08 2008