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A035013 One third of 9-factorial numbers.

%I A035013 #25 Dec 21 2022 04:47:09
%S A035013 1,12,252,7560,294840,14152320,806682240,53241027840,3993077088000,
%T A035013 335418475392000,31193918211456000,3181779657568512000,
%U A035013 353177541990104832000,42381305038812579840000,5467188350006822799360000,754471992300941546311680000,110907382868238407307816960000,17301551727445191540019445760000
%N A035013 One third of 9-factorial numbers.
%C A035013 E.g.f. is g.f. for A034171(n-1).
%H A035013 G. C. Greubel, <a href="/A035013/b035013.txt">Table of n, a(n) for n = 1..325</a>
%H A035013 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A035013 3*a(n) = (9*n-6)(!^9) := Product_{j=1..n} (9*j-6) = 3^n*A007559(n).
%F A035013 E.g.f.: (-1+(1-9*x)^(-1/3))/3.
%F A035013 From _G. C. Greubel_, Oct 18 2022: (Start)
%F A035013 a(n) = (1/3) * 9^n * Pochhammer(n, 1/3).
%F A035013 a(n) = (9*n-6)*a(n-1). (End)
%F A035013 From _Amiram Eldar_, Dec 21 2022: (Start)
%F A035013 a(n) = A144758(n)/3.
%F A035013 Sum_{n>=1} 1/a(n) = 3*(e/9^6)^(1/9)*(Gamma(1/3) - Gamma(1/3, 1/9)). (End)
%t A035013 s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 2*5!, 9}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 08 2008 *)
%t A035013 Table[9^n*Pochhammer[1/3, n]/3, {n, 40}] (* _G. C. Greubel_, Oct 18 2022 *)
%o A035013 (Magma) [n le 1 select 1 else (9*n-6)*Self(n-1): n in [1..40]]; // _G. C. Greubel_, Oct 18 2022
%o A035013 (SageMath) [9^n*rising_factorial(1/3,n)/3 for n in range(1,40)] # _G. C. Greubel_, Oct 18 2022
%Y A035013 Cf. A007559, A034171, A035012, A035013, A035017, A035018.
%Y A035013 Cf. A035020, A035021, A035022, A035023, A045756, A144758.
%K A035013 easy,nonn
%O A035013 1,2
%A A035013 _Wolfdieter Lang_
%E A035013 Terms a(15) onward added by _G. C. Greubel_, Oct 18 2022