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

1, 9, 405, 23085, 1454355, 96860043, 6683342967, 472607824095, 34027763334840, 2484026723443320, 183321172190117016, 13649094547609621464, 1023682091070721609800, 77248625487721376862600, 5859860019140007302005800, 446521333458468556412841960, 34158882009572844565582409940

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Armin Straub, Victor H. Moll, and Tewodros Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009), 31-41, eq (1.10).
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A007559, A034171, A035012, A035013, A035017, A035018, A035020, A035021, A035022, A035023, A045756, A256190.

Programs

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

Mathematica

CoefficientList[Series[1/Surd[1-81x,9],{x,0,20}],x] (* Harvey P. Dale, Mar 08 2018 *)
Table[9^(2*n)*Pochhammer[1/9, n]/n!, {n,0,40}] (* G. C. Greubel, Oct 19 2022 *)

SageMath

[9^(2*n)*rising_factorial(1/9,n)/factorial(n) for n in range(40)] # G. C. Greubel, Oct 19 2022

Formula

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

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

D-finite with recurrence: n*a(n) = 9*(9*n-8)*a(n-1). - R. J. Mathar, Jan 28 2020

a(n) = 9^(2*n) * Pochhammer(n, 1/9)/n!. - G. C. Greubel, Oct 19 2022

a(n) ~ 3^(4*n) * n^(-8/9) / Gamma(1/9). - Amiram Eldar, Aug 18 2025