%I #27 Dec 21 2022 04:45:59

%S 1,17,442,15470,680680,36076040,2236714480,158806728080,

%T 12704538246400,1130703903929600,110808982585100800,

%U 11856561136605785600,1375361091846271129600,171920136480783891200000,23037298288425041420800000,3294333655244780923174400000,500738715597206700322508800000

%N One eighth of 9-factorial numbers.

%H G. C. Greubel, <a href="/A035022/b035022.txt">Table of n, a(n) for n = 1..325</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%F 8*a(n) = (9*n-1)(!^9) := Product_{j=1..n} (9*j - 1).

%F a(n) = (9*n)!/(n!*2^4*3^(4*n)*5*7*A045756(n)*A035012(n)*A007559(n)*A035017(n) *A035018(n)*A034000(n) *A035021(n)).

%F E.g.f.: (-1+(1-9*x)^(-8/9))/8.

%F D-finite with recurrence: a(1) = 1, a(n) = (9*n - 1)*a(n-1) for n > 1. - _Georg Fischer_, Feb 15 2020

%F a(n) = (1/8) * 9^n * Pochhammer(n, 8/9). - _G. C. Greubel_, Oct 19 2022

%F From _Amiram Eldar_, Dec 21 2022: (Start)

%F a(n) = A049211(n)/8.

%F Sum_{n>=1} 1/a(n) = 8*(e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). (End)

%p f := gfun:-rectoproc({(9*n - 1)*a(n - 1) - a(n) = 0, a(1) = 1}, a(n), remember);

%p map(f, [$ (1 .. 20)]); # _Georg Fischer_, Feb 15 2020

%t Table[9^n*Pochhammer[8/9, n]/8, {n, 40}] (* _G. C. Greubel_, Oct 19 2022 *)

%o (Magma) [n le 1 select 1 else (9*n-1)*Self(n-1): n in [1..40]]; // _G. C. Greubel_, Oct 19 2022

%o (SageMath) [9^n*rising_factorial(8/9,n)/8 for n in range(1,40)] # _G. C. Greubel_, Oct 19 2022

%Y Cf. A007559, A034000, A034171, A035012, A035013, A035017, A049211.

%Y Cf. A035018, A035020, A035021, A035022, A035023, A045756.

%K easy,nonn

%O 1,2

%A _Wolfdieter Lang_