A106184 Expansion of 1/sqrt(1-4*x-8*x^2+32*x^3).

1, 2, 10, 28, 118, 380, 1508, 5240, 20326, 73836, 284396, 1061128, 4085820, 15500120, 59820040, 229366768, 887943046, 3428967500, 13315684764, 51678099304, 201246353492, 783890932488, 3060144292600, 11953056489104

Offset: 0

Author

Paul Barry, Apr 24 2005

Keywords

• nonn
• easy

Comments

In general, a(n)=sum{k=0..floor(n/2), C(2k,k)C(2(n-2k),n-2k)*r^k} has g.f. 1/sqrt(1-4x-4r*x^2+16r*x^3).

Gould's sequence (A001316) divides a. [g|a <=> g(n)|a(n) for all n]. A268433 is the termwise quotient. - Peter Luschny, Feb 24 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000984, A001316, A268433.

Programs

Maple

S:= series(1/sqrt(1-4*x-8*x^2+32*x^3),x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Feb 23 2016

Mathematica

CoefficientList[Series[1/Sqrt[1-4x-8x^2+32x^3],{x,0,30}],x] (* Harvey P. Dale, Sep 15 2011 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} C(2*k, k)*C(2(n-2*k), n-2*k)*2^k.

D-finite with recurrence: n*a(n)+2*(1-2*n)*a(n-1)+8*(1-n)*a(n-2)+16*(2*n-3)*a(n-3) = 0. - R. J. Mathar, Dec 08 2011

a(n) ~ 2^(2*n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 17 2012

G.f. g(x) satisfies (4*x-1)*(8*x^2-1)*g'(x) + (48*x^2-8*x-2)*g(x) = 0, from which Mathar's recurrence can be derived. - Robert Israel, Feb 23 2016

a(n) = C(2*n,n)*hypergeom([1/2,-n/2,-n/2+1/2],[-n/2+3/4,-n/2+1/4],1/2). - Peter Luschny, Feb 23 2016