A301271 Expansion of (1-16*x)^(1/8).

1, -2, -14, -140, -1610, -19964, -259532, -3485144, -47920730, -670890220, -9526641124, -136837208872, -1984139528644, -28998962341720, -426699017313880, -6315145456245424, -93937788661650682, -1403541077650545484, -21053116164758182260, -316904801216886322440

Offset: 0

Author

Seiichi Manyama, Jun 15 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..833

Crossrefs

Cf. A003557, A007947.

(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), this sequence (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), A303007 (b=240), A303055 (b=504), A305886 (b=1728).

Programs

PARI

N=20; x='x+O('x^N); Vec((1-16*x)^(1/8))

Formula

a(n) = 2^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.

a(n) = -sqrt(2-sqrt(2)) * Gamma(1/8) * Gamma(n-1/8) * 16^(n-1) / (Pi*Gamma(n+1)). - Vaclav Kotesovec, Jun 16 2018

a(n) ~ -2^(4*n-3) / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018

D-finite with recurrence: n*a(n) +2*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

a(n) = -2*A097184(n-1). - R. J. Mathar, Jan 20 2020