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

0,2

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

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

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A101500, A102880.

Sequence in context: A054936 A006443 A152103 * A032312 A032222 A212268

Adjacent sequences: A102876 A102877 A102878 * A102880 A102881 A102882

Keyword

easy,nonn

Author

Paul Barry, Jan 15 2005

Status

approved