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%I #8 Nov 05 2021 03:51:46
%S 1,1,3,12,53,250,1234,6295,32925,175616,951596,5223658,28987546,
%T 162349759,916502869,5209630108,29792226533,171284524184,989460348216,
%U 5740230703588,33429379234924,195361236443008,1145312096390408,6733896357727242,39697441350016170,234596104853541967
%N G.f. A(x) satisfies: A(x) = 1 + x*A(x)/(1 - 2*x*A(x) - x^2*A(x)^2).
%F G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Pell(k)*x^k*A(x)^k, where Pell = A000129.
%F G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - 2*x - x^2)/(1 - x - x^2)).
%F a(n) ~ sqrt((1 + 2^(1/3))*(4 + 7*2^(1/3))) * (2 + 3/2^(2/3) + 3/2^(1/3))^n / (3 * sqrt(Pi) * (2*n)^(3/2)). - _Vaclav Kotesovec_, Nov 05 2021
%t terms = 25; A[_] = 0; Do[A[x_] = 1 + x A[x]/(1 - 2 x A[x] - x^2 A[x]^2) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
%t terms = 26; A[_] = 0; Do[A[x_] = 1 + Sum[Fibonacci[k, 2] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
%t terms = 26; CoefficientList[1/x InverseSeries[Series[x (1 - 2 x - x^2)/(1 - x - x^2), {x, 0, terms}], x], x]
%Y Cf. A000129, A049124, A307411, A307413.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 07 2019
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