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A035024 Expansion of 1/(1-81*x)^(1/9), related to 9-factorial numbers A045756.

%I A035024 #33 Aug 18 2025 02:12:34
%S A035024 1,9,405,23085,1454355,96860043,6683342967,472607824095,
%T A035024 34027763334840,2484026723443320,183321172190117016,
%U A035024 13649094547609621464,1023682091070721609800,77248625487721376862600,5859860019140007302005800,446521333458468556412841960,34158882009572844565582409940
%N A035024 Expansion of 1/(1-81*x)^(1/9), related to 9-factorial numbers A045756.
%H A035024 Seiichi Manyama, <a href="/A035024/b035024.txt">Table of n, a(n) for n = 0..500</a>
%H A035024 Armin Straub, Victor H. Moll, and Tewodros Amdeberhan, <a href="http://dx.doi.org/10.4064/aa140-1-2">The p-adic valuation of k-central binomial coefficients</a>, Acta Arith. 140 (1) (2009), 31-41, eq (1.10).
%H A035024 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A035024 a(n) = 9^n*A045756(n)/n!, n >= 1, where A045756(n) = (9*n-8)(!^9) = Product_{j=1..n} (9*j - 8).
%F A035024 G.f.: (1-81*x)^(-1/9).
%F A035024 D-finite with recurrence: n*a(n) = 9*(9*n-8)*a(n-1). - _R. J. Mathar_, Jan 28 2020
%F A035024 a(n) = 9^(2*n) * Pochhammer(n, 1/9)/n!. - _G. C. Greubel_, Oct 19 2022
%F A035024 a(n) ~ 3^(4*n) * n^(-8/9) / Gamma(1/9). - _Amiram Eldar_, Aug 18 2025
%t A035024 CoefficientList[Series[1/Surd[1-81x,9],{x,0,20}],x] (* _Harvey P. Dale_, Mar 08 2018 *)
%t A035024 Table[9^(2*n)*Pochhammer[1/9, n]/n!, {n,0,40}] (* _G. C. Greubel_, Oct 19 2022 *)
%o A035024 (Magma) [n le 1 select 1 else 9*(9*n-17)*Self(n-1)/(n-1): n in [1..40]]; // _G. C. Greubel_, Oct 19 2022
%o A035024 (SageMath) [9^(2*n)*rising_factorial(1/9,n)/factorial(n) for n in range(40)] # _G. C. Greubel_, Oct 19 2022
%Y A035024 Cf. A007559, A034171, A035012, A035013, A035017, A035018, A035020, A035021, A035022, A035023, A045756, A256190.
%K A035024 easy,nonn
%O A035024 0,2
%A A035024 _Wolfdieter Lang_