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%I #17 Jul 20 2023 15:46:10
%S 1,1,0,-2,2,6,-18,-6,114,-146,-490,1794,266,-12986,20958,56778,
%T -255774,39390,1853478,-3687918,-7441158,42252726,-20345490,
%U -293463462,708206802,1002083406,-7527677898,6140678434,48978210794,-142206136026,-127715768578
%N Expansion of (1+5x+sqrt(1+2x+9x^2))/(2(1+2x)).
%C Hankel transform is A186196. Hankel transform of a(n+1) is (-2)^C(n+1,2).
%F D-finite with recurrence n*a(n) +(4*n-3)*a(n-1) +(13*n-33)*a(n-2) +18*(n-3)*a(n-3)=0. - _R. J. Mathar_, Feb 13 2015
%F From _Peter Bala_, Nov 08 2022: (Start)
%F O.g.f. A(x) = 1 + series reversion of x*(1 + x)/((1 - x)*(1 + 2*x)).
%F The g.f. satisfies the differential equation (1 + 4*x + 13*x^2 + 18*x^3)*A'(x) + (1 - 7*x)*A(x) + (2*x - 2) = 0 with A(0) = 1. Mathar's recurrence above follows from this. (End)
%t CoefficientList[Series[(1+5x+Sqrt[1+2x+9x^2])/(2(1+2x)),{x,0,30}],x] (* _Harvey P. Dale_, Dec 17 2021 *)
%Y Cf. A114710, A186196.
%K sign,easy
%O 0,4
%A _Paul Barry_, Feb 14 2011
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