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%I #30 Apr 01 2020 12:17:34
%S 1,1,5,9,37,81,301,729,2549,6561,22045,59049,193029,531441,1703469,
%T 4782969,15111573,43046721,134539837,387420489,1200901157,3486784401,
%U 10739313997,31381059609,96172251061,282429536481,862142190941,2541865828329,7734936371269,22876792454961,69439155241581
%N G.f.: 1/sqrt(1-10*x^2 + x^4/(1-8*x^2)) + x/(1-9*x^2).
%F Conjecture D-finite with recurrence: n*a(n) +(n-1)*a(n-1) +(24-17*n)*a(n-2) +(41-17*n)*a(n-3) +72*(n-3)*a(n-4) +72*(n-4)*a(n-5)=0. - _R. J. Mathar_, Nov 17 2011
%F G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x) + 9*x^2*A(x)^2). - _Paul D. Hanna_, Nov 18 2014
%F Let G(x) = g.f. of A200375, then g.f. A(x) satisfies:
%F (1) A(x) = x/Series_Reversion(x*G(x)),
%F (2) A(x) = G(x/A(x)) and G(x) = A(x*G(x)),
%F where A200375(n) = A000108(n)*A001045(n), the product of Catalan and Jacobsthal numbers.
%F a(n) ~ 3^(n-1). - _Vaclav Kotesovec_, Jun 29 2013
%e G.f.: A(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...
%e The g.f. of A200375(n) = A000108(n)*A001045(n) begins:
%e G(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 +...
%e where A(x) = G(x/A(x)) and G(x) = A(x*G(x)).
%t CoefficientList[Series[1/Sqrt[1-10x^2+x^4/(1-8x^2)]+x/(1-9x^2),{x,0,30}], x] (* _Harvey P. Dale_, Nov 19 2011 *)
%o (PARI) {a(n)=polcoeff(1/sqrt(1-10*x^2 + x^4/(1-8*x^2 +x*O(x^n))) + x/(1-9*x^2 +x*O(x^n)),n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=local(G=sum(m=0,n,binomial(2*m, m)/(m+1)*polcoeff(1/(1-x-2*x^2+x*O(x^m)), m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A200375, A098615, A098617.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 16 2011
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