A115967 Expansion of 1/(2*sqrt(1-2*x-3*x^2) - 1).

1, 2, 8, 28, 104, 384, 1428, 5316, 19820, 73948, 276044, 1030796, 3850048, 14382248, 53732172, 200759004, 750134520, 2802980640, 10474015164, 39139487292, 146259311592, 546558514368, 2042458815324, 7632600834924, 28522903136796

Offset

0,2

Comments

Row sums of number triangle A116392.

Links

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

Formula

a(n) = Sum_{k=0..n} A116392(n,k).

G.f.: A(x)/(2 - A(x)) where A(x) is the g.f. of the central trinomial coefficients A002426.

G.f.: (1 + 2*sqrt(1-2*x-3*x^2))/(3-8*x-12*x^2).

Hankel transform is A000302, A000302(n)=4^n. - Philippe Deléham, Jun 22 2007

G.f.: 1/(2*sqrt(1-2*x-3*x^2) - 1) = 1/(1 - 2*x/G(0)); G(k)= 1 - 2*x/(1 + x/(1 + x/(1 - 2*x/(1 - x/(2 - x/G(k+1)))))); (continued fraction, 6-step). - Sergei N. Gladkovskii, Feb 27 2012

Conjecture: 3*n*a(n) + (-14*n+9)*a(n-1) + (-5*n+3)*a(n-2) + 12*(4*n-9)* a(n-3) + 36*(n-3)*a(n-4) = 0. - R. J. Mathar, Nov 15 2012

a(n) ~ (1/9 + 2/(9*sqrt(13))) * (4+2*sqrt(13))^n / 3^(n-1). - Vaclav Kotesovec, Feb 08 2014

Mathematica

CoefficientList[ Series[1/(2 Sqrt[1-2x-3x^2]-1), {x, 0, 30}], x] (* Robert G. Wilson v, Feb 28 2012 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1 + 2*sqrt(1-2*x-3*x^2))/(3-8*x-12*x^2)) \\ G. C. Greubel, May 06 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 + 2*Sqrt(1-2*x-3*x^2))/(3-8*x-12*x^2) )); // G. C. Greubel, May 06 2019
(Sage) ((1 + 2*sqrt(1-2*x-3*x^2))/(3-8*x-12*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 06 2019

Crossrefs

Sequence in context: A056711 A114590 A133592 * A357641 A150714 A292668

Adjacent sequences: A115964 A115965 A115966 * A115968 A115969 A115970

Keyword

easy,nonn

Author

Paul Barry, Feb 03 2006

Extensions

Entry revised by N. J. A. Sloane, Apr 10 2006

Status

approved