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A035021 One seventh of 9-factorial numbers.

%I A035021 #26 Dec 21 2022 04:45:56
%S A035021 1,16,400,13600,584800,30409600,1854985600,129848992000,
%T A035021 10258070368000,902710192384000,87562888661248000,9281666198092288000,
%U A035021 1067391612780613120000,132356559984796026880000,17603422477977871575040000,2499685991872857763655680000,377452584772801522312007680000
%N A035021 One seventh of 9-factorial numbers.
%H A035021 G. C. Greubel, <a href="/A035021/b035021.txt">Table of n, a(n) for n = 1..325</a>
%H A035021 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A035021 7*a(n) = (9*n-2)(!^9) := Product_{j=1..n} (9*j-2).
%F A035021 E.g.f.: (-1+(1-9*x)^(-7/9))/7.
%F A035021 D-finite with recurrence: a(1) = 1, a(n) = (9*n - 2)*a(n-1) for n > 1. - _Georg Fischer_, Feb 15 2020
%F A035021 a(n) = (1/7) * 9^n * Pochhammer(n, 7/9). - _G. C. Greubel_, Oct 19 2022
%F A035021 From _Amiram Eldar_, Dec 21 2022: (Start)
%F A035021 a(n) = A147631(n+1)/7.
%F A035021 Sum_{n>=1} 1/a(n) = 7*(e/9^2)^(1/9)*(Gamma(7/9) - Gamma(7/9, 1/9)). (End)
%p A035021 f := gfun:-rectoproc({(9*n - 2)*a(n - 1) - a(n) = 0, a(1) = 1}, a(n), remember);
%p A035021 map(f, [$ (1 .. 20)]); # _Georg Fischer_, Feb 15 2020
%t A035021 Table[9^n*Pochhammer[7/9, n]/7, {n, 40}] (* _G. C. Greubel_, Oct 19 2022 *)
%o A035021 (Magma) [n le 1 select 1 else (9*n-2)*Self(n-1): n in [1..40]]; // _G. C. Greubel_, Oct 19 2022
%o A035021 (SageMath) [9^n*rising_factorial(7/9,n)/7 for n in range(1,40)] # _G. C. Greubel_, Oct 19 2022
%Y A035021 Cf. A007559, A034171, A035012, A035013, A035017, A035018.
%Y A035021 Cf. A035020, A035021, A035022, A035023, A045756, A147631.
%K A035021 easy,nonn
%O A035021 1,2
%A A035021 _Wolfdieter Lang_
%E A035021 Terms a(15) onward added by _G. C. Greubel_, Oct 19 2022