A162543 - OEIS

%I #15 Sep 08 2022 08:45:46

%S 1,2,5,18,73,312,1391,6406,30235,145478,710951,3519248,17608681,

%T 88914250,452512229,2318774506,11953427329,61948592936,322570037543,

%U 1686777086942,8854240330363,46638995523598,246443050810895

%N A Chebyshev transform of the large Schroeder numbers A006318.

%C Hankel transform is the Somos-4 variant A162546.

%H Fung Lam, <a href="/A162543/b162543.txt">Table of n, a(n) for n = 0..1335</a>

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

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

%F 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);

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

%F 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

%t 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 *)

%o (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

%o (Magma) m:=30; R<x>:=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

%o (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

%o (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

%Y Cf. A162548.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 05 2009