A115970 Expansion of 1/(4*sqrt(1-4*x) - 3).

1, 8, 72, 656, 5992, 54768, 500688, 4577568, 41851560, 382641200, 3498428272, 31985610720, 292439802256, 2673735097184, 24445577182368, 223502416896576, 2043450657688872, 18682977401318064, 170815793235313968

Offset: 0

Author

Paul Barry, Feb 03 2006

Keywords

• easy
• nonn

Comments

The g.f. is A(x)^2/(2*A(x)-A(x)^2) where A(x) is the g.f. of A076035.

The Hankel transform of this sequence is 8^n = [1, 8, 64, 512, 4096, ...]; the Hankel transform of the aerated sequence with g.f. 1/(1-8*x^2*c(x^2)) is also 8^n. - Philippe Deléham, Feb 13 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Programs

Magma

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( 1/(4*Sqrt(1-4*x)-3) )); // G. C. Greubel, May 05 2019

Mathematica

CoefficientList[Series[1/(4*Sqrt[1-4*x]-3), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

PARI

my(x='x+O('x^20)); Vec(1/(4*sqrt(1-4*x)-3)) \\ G. C. Greubel, May 05 2019

Sage

(1/(4*sqrt(1-4*x)-3)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 05 2019

Formula

G.f.: 1/(1-8*x*c(x)), where c(x) is the g.f. of A000108.

a(n) = Sum_{k=0..n} A106566(n, k)*8^k.

From Philippe Deléham, Feb 13 2007: (Start)

a(n) = (64*a(n-1) - 8*A000108(n-1))/7.

a(n) = Sum_{k=0..n} A039599(n,k)*7^k.

a(n) = Sum_{k=0..n} A106566(n,k)*8^k. (End)

D-finite with recurrence: 7*n*a(n) = 2*(46*n-21)*a(n-1) - 128*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 3*2^(6*n+1)/7^(n+1). - Vaclav Kotesovec, Oct 19 2012