A307490 - OEIS

%I #7 Sep 16 2021 13:56:16

%S 1,1,3,10,39,161,700,3144,14495,68167,325787,1577654,7724612,38176620,

%T 190196440,954182528,4816268367,24441691827,124633707865,638272397350,

%U 3281390284623,16929034639153,87617665434328,454796186669800,2367025393846724,12349738834935876

%N G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x) - x^2*A(x)^2/(1 - x^2*A(x)^2/(1 - x^2*A(x)^2/(1 - ...)))), a continued fraction.

%F G.f. A(x) satisfies: A(x) = 2/(1 - 2*x*A(x) + sqrt(1 - 4*x^2*A(x)^2)).

%F G.f. A(x) satisfies: A(x) = Sum_{k>=0} binomial(k,floor(k/2))*x^k*A(x)^k.

%F G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - 2*x + sqrt(1 - 4*x^2))/2).

%F a(n) ~ 1/(n^(3/2)*sqrt(Pi*(103/6 + (1/6)*sqrt(36037)*cos((1/3)*(4*Pi + arccos(6832781/(36037*sqrt(36037))))))) * (-7/48 + (1/24)*sqrt(203/2) * cos((1/3)*arccos(-(1849/(203*sqrt(406))))))^n). - _Vaclav Kotesovec_, Sep 16 2021

%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 161*x^5 + 700*x^6 + 3144*x^7 + 14495*x^8 + 68167*x^9 + 325787*x^10 + ...

%t terms = 26; CoefficientList[1/x InverseSeries[Series[x (1 - 2 x + Sqrt[1 - 4 x^2])/2, {x, 0, terms}], x], x]

%t terms = 26; A[_] = 0; Do[A[x_] = 1/(1 - x A[x] + ContinuedFractionK[-x^2 A[x]^2, 1, {k, 1, j}]) + O[x]^j, {j, 1,terms}]; CoefficientList[A[x], x]

%t terms = 25; A[_] = 1; Do[A[x_] = 2/(1 - 2 x A[x] + Sqrt[1 - 4 x^2 A[x]^2]) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]

%t terms = 26; A[_] = 1; Do[A[x_] = Sum[Binomial[k, Floor[k/2]] x^k A[x]^k, {k, 0, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

%Y Cf. A001405, A001764, A078531, A307489.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 10 2019