A102879 - OEIS

%I #20 Nov 02 2023 10:07:58

%S 1,2,4,14,48,162,556,1934,6784,23954,85044,303294,1085712,3898962,

%T 14040156,50678814,183309312,664263714,2411050084,8764098158,

%U 31899231088,116244082178,424064770188,1548543412398,5659898710912

%N A Chebyshev transform of the first kind of the central binomial numbers.

%C Image of 1/sqrt(1-4x) under the mapping g(x)->((1-x^2)/(1+x^2))*g(x/(1+x^2)).

%H Robert Israel, <a href="/A102879/b102879.txt">Table of n, a(n) for n = 0..1749</a>

%F G.f.: (1-x^2)/sqrt(1-4*x+2*x^2-4*x^3+x^4).

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

%F Conjecture: n*(n-3)*a(n) - 2*(2*n-1)*(n-3)*a(n-1) + 2*(2-4*n+n^2)*a(n-2) - 2*(n-1)*(2*n-7)*a(n-3) + (n-1)*(n-4)*a(n-4) = 0. - _R. J. Mathar_, Nov 09 2012

%F Conjecture verified using the differential equation x*(x^2+1)*(x^2-4*x+1)*g'' + (4*x^4-10*x^3+2*x^2+2*x-2)*g' + 4*(x^2-x+1)*g = 0 satisfied by the g.f. - _Robert Israel_, Aug 28 2018

%F a(n) ~ 3^(1/4) * (2 + sqrt(3))^n / sqrt(2*Pi*n). - _Vaclav Kotesovec_, Nov 02 2023

%p f:= gfun:-rectoproc({n*(n-3)*a(n) -2*(2*n-1)*(n-3)*a(n-1) +2*(2-4*n+n^2)*a(n-2) -2*(n-1)*(2*n-7)*a(n-3) +(n-1)*(n-4)*a(n-4),a(0)=1,a(1)=2,a(2)=4,a(3)=14},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, Aug 28 2018

%t CoefficientList[Series[(1-x^2)/Sqrt[1-4*x+2*x^2-4*x^3+x^4], {x,0,30}],x] (* _G. C. Greubel_, Mar 31 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec((1-x^2)/sqrt(1-4*x+2*x^2-4*x^3+x^4)) \\ _G. C. Greubel_, Mar 31 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-x^2)/Sqrt(1-4*x+2*x^2-4*x^3+x^4) )); // _G. C. Greubel_, Mar 31 2019

%o (Sage) ((1-x^2)/sqrt(1-4*x+2*x^2-4*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Mar 31 2019

%Y Cf. A101500, A102880.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jan 15 2005