A106186 Expansion of 1/sqrt(1-4x+4x^2-16x^3).

1, 2, 4, 16, 64, 224, 800, 3008, 11392, 43008, 163328, 624640, 2397696, 9227264, 35608576, 137764864, 534102016, 2074394624, 8069922816, 31440338944, 122652655616, 479052693504, 1873097261056, 7331070869504, 28718945140736

Offset

0,2

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).

Links

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

Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Formula

a(n)=sum{k=0..floor(n/2), C(2k, k)C(2(n-2k), n-2k)*(-1)^k}.

D-finite with recurrence: n*a(n)+2(1-2n)*a(n-1) +4(n-1)*a(n-2)+8(3-2n)*a(n-3)=0. - R. J. Mathar, Dec 08 2011

a(n) ~ 2^(2*n+1)/sqrt(5*Pi*n). - Vaclav Kotesovec, Feb 01 2014

Mathematica

CoefficientList[Series[1/Sqrt[1-4x+4x^2-16x^3], {x, 0, 30}], x] Harvey P. Dale, May 17 2012

Crossrefs

Sequence in context: A127588 A187822 A092585 * A155543 A151371 A001900

Adjacent sequences: A106183 A106184 A106185 * A106187 A106188 A106189

Keyword

easy,nonn

Author

Paul Barry, Apr 24 2005

Status

approved