A162548 A Chebyshev transform of the little Schroeder numbers A001003.

1, 1, 2, 9, 37, 156, 695, 3203, 15118, 72739, 355475, 1759624, 8804341, 44457125, 226256114, 1159387253, 5976713665, 30974296468, 161285018771, 843388543471, 4427120165182, 23319497761799, 123221525405447, 652989260163472

Offset: 0

Author

Paul Barry, Jul 05 2009

Keywords

• easy
• nonn

Comments

Hankel transform is Somos-4 variant A162546.

Links

• Fung Lam, Table of n, a(n) for n = 0..1335

Crossrefs

Cf. A001003, A162543.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+x+x^2 -Sqrt(1-6*x+3*x^2-6*x^3+x^4))/(4*x*(1+x^2)) )); // G. C. Greubel, Feb 26 2019

Mathematica

CoefficientList[Series[(1+x+x^2 -Sqrt[1-6*x+3*x^2-6*x^3+x^4])/(4*x*(1+x^2)), {x,0,30}], x] (* G. C. Greubel, Feb 26 2019 *)

PARI

my(x='x+O('x^30)); Vec((1+x+x^2 -sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 4*x*(1+x^2))) \\ G. C. Greubel, Feb 26 2019

Sage

((1+x+x^2 -sqrt(1-6*x+3*x^2-6*x^3+x^4))/(4*x*(1+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

Formula

G.f.: (1/(1+x^2))*s(x/(1+x^2)), s(x) the g.f. of A001003.

G.f.: (1+x+x^2 - sqrt(1-6*x+3*x^2-6*x^3+x^4))/(4*x*(1+x^2)).

G.f.: 1/(1+x^2-x/(1-2*x/(1+x^2-x/(1-2*x/(1+x^2-x/(1-2*x/(1+x+x^2/(1-... (continued fraction);

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(n-k,k)*A001003(n-2k).

Conjecture: (n+1)*a(n) +3*(-2*n+1)*a(n-1) +(4*n-5)*a(n-2) +12*(-n+2)*a(n-3) +(4*n-11)*a(n-4) +3*(-2*n+7)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Nov 15 2012. (Formula verified and used for computations. - Fung Lam, Feb 19 2014)