A034977 Expansion of 1/(1-64*x)^(1/8), related to octo-factorial numbers A045755.

1, 8, 288, 13056, 652800, 34467840, 1884241920, 105517547520, 6014500208640, 347504456499200, 20294260259553280, 1195516422562775040, 70933974405391319040, 4234212626044897198080, 254052757562693831884800, 15310912855778348268257280, 926310227774590070229565440

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. A004993, A034855, A045755, A203142.

Programs

Magma

[n le 1 select 8^(n-1) else 8*(8*n-15)*Self(n-1)/(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022

Mathematica

CoefficientList[Series[1/(1-64x)^(1/8),{x,0,30}],x] (* Harvey P. Dale, May 20 2011 *)

SageMath

[2^(6*n)*rising_factorial(1/8,n)/factorial(n) for n in range(40)] # G. C. Greubel, Oct 21 2022

Formula

a(n) = 8^n*A045755(n)/n!, n >= 1, where A045755(n) = (8*n-7)!^8 = Product_{j=1..n} (8*j-7).

G.f.: (1-64*x)^(-1/8).

D-finite with recurrence: n*a(n) = 8*(8*n-7)*a(n-1). - R. J. Mathar, Jan 28 2020

a(n) ~ 2^(6*n) * n^(-7/8) / Gamma(1/8). - Amiram Eldar, Aug 18 2025

Extensions

a(11) corrected by Harvey P. Dale, May 20 2011