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

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

A. Straub, V. H. Moll, and T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10)

Formula

a(n) = 9^n*A045756(n)/n!, n >= 1; 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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A063156 A058850 A151632 * A179433 A091061 A024123

Adjacent sequences: A035021 A035022 A035023 * A035025 A035026 A035027

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved