A102879 A Chebyshev transform of the first kind of the central binomial numbers.

1, 2, 4, 14, 48, 162, 556, 1934, 6784, 23954, 85044, 303294, 1085712, 3898962, 14040156, 50678814, 183309312, 664263714, 2411050084, 8764098158, 31899231088, 116244082178, 424064770188, 1548543412398, 5659898710912

Offset: 0

Author

Paul Barry, Jan 15 2005

Keywords

• easy
• nonn

Comments

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

Links

• Robert Israel, Table of n, a(n) for n = 0..1749

Crossrefs

Cf. A101500, A102880.

Programs

Magma

R:=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

Maple

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):
map(f, [$0..30]); # Robert Israel, Aug 28 2018

Mathematica

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 *)

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

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

Formula

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

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

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

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

a(n) ~ 3^(1/4) * (2 + sqrt(3))^n / sqrt(2*Pi*n). - Vaclav Kotesovec, Nov 02 2023