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Expansion of 1/(1-81*x)^(1/9), related to 9-factorial numbers A045756.
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%I #23 Mar 20 2024 16:36:52

%S 1,9,405,23085,1454355,96860043,6683342967,472607824095,

%T 34027763334840,2484026723443320,183321172190117016,

%U 13649094547609621464,1023682091070721609800,77248625487721376862600

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

%H Seiichi Manyama, <a href="/A035024/b035024.txt">Table of n, a(n) for n = 0..500</a>

%H A. Straub, V. H. Moll, and T. 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)

%F a(n) = 9^n*A045756(n)/n!, n >= 1; A045756(n)= (9*n-8)(!^9) := Product_{j=1..n} (9*j - 8).

%F G.f.: (1-81*x)^(-1/9).

%F D-finite with recurrence: n*a(n) = 9*(9*n-8)*a(n-1). - _R. J. Mathar_, Jan 28 2020

%F a(n) = 9^(2*n) * Pochhammer(n, 1/9)/n!. - _G. C. Greubel_, Oct 19 2022

%t CoefficientList[Series[1/Surd[1-81x,9],{x,0,20}],x] (* _Harvey P. Dale_, Mar 08 2018 *)

%t Table[9^(2*n)*Pochhammer[1/9, n]/n!, {n,0,40}] (* _G. C. Greubel_, Oct 19 2022 *)

%o (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 (SageMath) [9^(2*n)*rising_factorial(1/9,n)/factorial(n) for n in range(40)] # _G. C. Greubel_, Oct 19 2022

%Y Cf. A007559, A034171, A035012, A035013, A035017, A035018, A035020, A035021, A035022, A035023, A045756.

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_