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%I #12 Sep 08 2022 08:45:45
%S 1,1,3,5,13,27,69,161,415,1033,2701,6983,18521,49041,131723,354493,
%T 962381,2620675,7178285,19724513,54430023,150641937,418294813,
%U 1164528399,3250685297,9094701729,25501672595,71649158709,201687341901
%N A transform of the large Schroeder numbers.
%C Hankel transform is A060656(n+1).
%H G. C. Greubel, <a href="/A160823/b160823.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: 1/(1-x-2x^2/(1-x^2/(1-x-2x^2/(1-x^2/(1-x-2x^2/(1-x^2/(1-...))))))) (continued fraction);
%F G.f.: (1-x-x^2-sqrt(1-2*x-5*x^2+6*x^3+x^4))/(2*x^2*(1-x)).
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*A006318(k).
%F Conjecture: (n+2)*a(n) -3*(n+1)*a(n-1) +3(2-n)*a(n-2) +(11*n-20)*a(n-3) +(11-5*n)*a(n-4) + (4-n)*a(n-5)=0. - _R. J. Mathar_, Nov 16 2011
%F a(n) ~ sqrt((-36 + 63*sqrt(2) + sqrt(8666 - 4936*sqrt(2)))/8) * ((1 + sqrt(13 + 8*sqrt(2)))/2)^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, May 01 2018
%e G.f. = 1 + x + 3*x^2 + 5*x^3 + 13*x^4 + 27*x^5 + 69*x^6 + 161*x^7 + ...
%t CoefficientList[Series[(1-x-x^2-Sqrt[1-2*x-5*x^2+6*x^3+x^4])/(2*x^2*(1- x)), {x, 0, 50}], x] (* _G. C. Greubel_, Apr 30 2018 *)
%o (PARI) x='x+O('x^50); Vec((1-x-x^2-sqrt(1-2*x-5*x^2+6*x^3+x^4))/(2*x^2*(1-x))) \\ _G. C. Greubel_, Apr 30 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x-x^2-Sqrt(1-2*x-5*x^2+6*x^3+x^4))/(2*x^2*(1-x)))); // _G. C. Greubel_, Apr 30 2018
%K easy,nonn
%O 0,3
%A _Paul Barry_, May 27 2009
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