|
%I #31 Sep 08 2022 08:45:24
%S 1,2,8,28,104,384,1428,5316,19820,73948,276044,1030796,3850048,
%T 14382248,53732172,200759004,750134520,2802980640,10474015164,
%U 39139487292,146259311592,546558514368,2042458815324,7632600834924,28522903136796
%N Expansion of 1/(2*sqrt(1-2*x-3*x^2) - 1).
%C Row sums of number triangle A116392.
%H Vincenzo Librandi, <a href="/A115967/b115967.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..n} A116392(n,k).
%F G.f.: A(x)/(2 - A(x)) where A(x) is the g.f. of the central trinomial coefficients A002426.
%F G.f.: (1 + 2*sqrt(1-2*x-3*x^2))/(3-8*x-12*x^2).
%F Hankel transform is A000302, A000302(n)=4^n. - _Philippe Deléham_, Jun 22 2007
%F 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
%F 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
%F a(n) ~ (1/9 + 2/(9*sqrt(13))) * (4+2*sqrt(13))^n / 3^(n-1). - _Vaclav Kotesovec_, Feb 08 2014
%t CoefficientList[ Series[1/(2 Sqrt[1-2x-3x^2]-1), {x, 0, 30}], x] (* _Robert G. Wilson v_, Feb 28 2012 *)
%o (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
%o (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
%o (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
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 03 2006
%E Entry revised by _N. J. A. Sloane_, Apr 10 2006
| 