A224881 Expansion of 1/(1 - 16*x)^(1/8).

1, 2, 18, 204, 2550, 33660, 460020, 6440280, 91773990, 1325624300, 19354114780, 285033326760, 4227994346940, 63094684869720, 946420273045800, 14259398780556720, 215673406555920390, 3273161111260438860, 49824785804742235980, 760483572809223601800

Offset: 0

Author

Paul D. Hanna, Jul 23 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 204*x^3 + 2550*x^4 + 33660*x^5 + ...
where
A(x)^8 = 1 + 16*x + 256*x^2 + 4096*x^3 + 65536*x^4 + ... + 16^n*x^n + ...
Also,
A(x)^4 = 1 + 8*x + 96*x^2 + 1280*x^3 + 17920*x^4 + 258048*x^5 + ... + 4^n*A000984(n)*x^n + ...
A(x)^2 = 1 + 4*x + 40*x^2 + 480*x^3 + 6240*x^4 + 84864*x^5 + ... + 2^n*A004981(n)*x^n + ...
		

Links

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

Crossrefs

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), this sequence (b=16), A034688 (b=25), A298799 (b=27), A004993 (b=36), A034835 (b=49).

Cf. A301271.

Programs

GAP

List([0..20],n->(2^n/Factorial(n))*Product([0..n-1],k->8*k+1)); # Muniru A Asiru, Jun 23 2018

Maple

seq(coeff(series(1/(1-16*x)^(1/8), x,50),x,n+1),n=0..20); # Muniru A Asiru, Jun 23 2018

Mathematica

CoefficientList[Series[1/(1-16*x)^(1/8), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 24 2013 *)

PARI

{a(n)=polcoeff(1/(1-16*x +x*O(x^n))^(1/8),n)}
for(n=0,30,print1(a(n),", "))

PARI

{a(n)=(2^n/n!)*prod(k=0,n-1,8*k + 1)}
for(n=0,30,print1(a(n),", "))

Formula

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

a(n) ~ 16^n/(GAMMA(1/8)*n^(7/8)). - Vaclav Kotesovec, Jul 24 2013