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A162548 A Chebyshev transform of the little Schroeder numbers A001003.

%I A162548 #15 Sep 08 2022 08:45:46
%S A162548 1,1,2,9,37,156,695,3203,15118,72739,355475,1759624,8804341,44457125,
%T A162548 226256114,1159387253,5976713665,30974296468,161285018771,
%U A162548 843388543471,4427120165182,23319497761799,123221525405447,652989260163472
%N A162548 A Chebyshev transform of the little Schroeder numbers A001003.
%C A162548 Hankel transform is Somos-4 variant A162546.
%H A162548 Fung Lam, <a href="/A162548/b162548.txt">Table of n, a(n) for n = 0..1335</a>
%F A162548 G.f.: (1/(1+x^2))*s(x/(1+x^2)), s(x) the g.f. of A001003.
%F A162548 G.f.: (1+x+x^2 - sqrt(1-6*x+3*x^2-6*x^3+x^4))/(4*x*(1+x^2)).
%F A162548 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);
%F A162548 a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(n-k,k)*A001003(n-2k).
%F A162548 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)
%t A162548 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 *)
%o A162548 (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
%o A162548 (Magma) R<x>:=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
%o A162548 (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
%Y A162548 Cf. A001003, A162543.
%K A162548 easy,nonn
%O A162548 0,3
%A A162548 _Paul Barry_, Jul 05 2009