A162543 A Chebyshev transform of the large Schroeder numbers A006318.

1, 2, 5, 18, 73, 312, 1391, 6406, 30235, 145478, 710951, 3519248, 17608681, 88914250, 452512229, 2318774506, 11953427329, 61948592936, 322570037543, 1686777086942, 8854240330363, 46638995523598, 246443050810895

Offset: 0

Author

Paul Barry, Jul 05 2009

Keywords

• easy
• nonn

Comments

Hankel transform is the Somos-4 variant A162546.

Links

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

Crossrefs

Cf. A162548.

Programs

GAP

a:=[2,5,18,73,312,1391];; for n in [7..30] do 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])/(n+1); od; Concatenation([1], a); # G. C. Greubel, Feb 23 2019

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-x+x^2 - Sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2)) )); // G. C. Greubel, Feb 23 2019

Mathematica

CoefficientList[Series[(1-x+x^2 - Sqrt[1-6*x+3*x^2-6*x^3+x^4])/(2*x*(1+x^2)), {n,0,30}], x] (* G. C. Greubel, Feb 23 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))/( 2*x*(1+x^2))) \\ G. C. Greubel, Feb 23 2019

Sage

((1-x+x^2 -sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 23 2019

Formula

G.f.: (1/(1+x^2))*S(x/(1+x^2)), S(x) the g.f. of A006318;

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

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

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*A006318(n-2*k).

Recurrence: (n+1)*a(n) = (5-n)*a(n-6) + 3*(2*n-7)*a(n-5) + (11-4*n)*a(n-4) + 12*(n-2)*a(n-3) + (5-4*n)*a(n-2) + 3*(2*n-1)*a(n-1), n>=6. - Fung Lam, Feb 19 2014