A179190 Coefficient [x^n] of the Maclaurin series for 2 - sqrt(1 - 4*x - 4*x^2).

1, 2, 4, 8, 24, 80, 288, 1088, 4256, 17088, 70016, 291584, 1230592, 5251584, 22623232, 98248704, 429677056, 1890700288, 8364824576, 37186449408, 166030266368, 744180244480, 3347321831424, 15104525959168, 68357598756864

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..151

Formula

G.f.: 2 - sqrt(1 - 4*x - 4*x^2).

a(n) = 4*A071356(n-2), n >= 2. - R. J. Mathar, Jul 08 2010

a(n) = Sum_{k=0..floor(n/2)} (2*n - 2*k - 3)!! *2^(n-k)/(k!*(n-2k)!), n > 0. - R. J. Mathar, Jul 11 2011

a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 26 2019

D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1) +4*(-n+3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020

Example

The Maclaurin series is 1 + 2*x + 4*x^2 + 8*x^3 + 24*x^4 + ...

Maple

A179190 := proc(n) if n = 0 then 1; else add( doublefactorial(2*n-2*k-3) *2^(n-k) / k! / (n-2*k)!, k=0..floor(n/2)) ; end if; end proc: # R. J. Mathar, Jul 11 2011

Mathematica

Table[SeriesCoefficient[Series[2-Sqrt[1-4*t-4*t^2], {t, 0, n}], n], {n, 0, 30}] (* G. C. Greubel, Jan 25 2019 *)

Prog

(Maxima) makelist(coeff(taylor(2-sqrt(1-4*x-4*x^2), x, 0, n), x, n), n, 0, 24); // Bruno Berselli, Jul 04 2011

Crossrefs

Cf. A178693, A178694, A179191.

Sequence in context: A264557 A067646 A152875 * A291482 A065654 A002908

Adjacent sequences: A179187 A179188 A179189 * A179191 A179192 A179193

Keyword

nonn

Author

Clark Kimberling, Jul 01 2010

Status

approved