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
Links
- Robert Israel, Table of n, a(n) for n = 0..1749
Programs
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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
Comments