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%I #31 Feb 10 2024 14:08:09
%S 1,2,7,34,195,1225,8146,56336,401005,2918308,21614216,162385693,
%T 1234515922,9479336440,73410868630,572719097908,4496923141241,
%U 35509806367132,281814387290431,2246608404455588,17982234787607464,144458551104066553,1164342291135424494
%N G.f. satisfies A(x) = Sum_{n>=0} x^n * (1 - A(x)^(2*n+2))/(1 - A(x)^2).
%C Compare to g.f. B(x) of A007317 (binomial transform of Catalan numbers):
%C B(x) = Sum_{n>=0} x^n * (1 - B(x)^(n+1))/(1 - B(x)).
%H Seiichi Manyama, <a href="/A199475/b199475.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. satisfies: A(x) = 1/((1-x)*(1 - x*A(x)^2)).
%F G.f.: A(x) = (1/x)*Series_Reversion( 2*x^2/(1+x^2 - sqrt(1-4*x-2*x^2+x^4)) ).
%F G.f. satisfies: A(x) = G(x*A(x)) and G(x) = A(x/G(x)) = g.f. of A171199, where G(x) = exp( Sum_{n>=1} [G(x)^n + G(x)^-n]*x^n/n ).
%F a(n) = 1 + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1). - _Ilya Gutkovskiy_, Jul 25 2021
%F a(n) ~ sqrt(387 + 35*sqrt(129)) * (35 + 3*sqrt(129))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - _Vaclav Kotesovec_, Jul 30 2021
%F a(n) = Sum_{k=0..n} binomial(n+k,n-k) * binomial(3*k,k)/(2*k+1). - _Seiichi Manyama_, Oct 03 2023
%F D-finite with recurrence 2*n*(2*n+1)*a(n) +3*(-13*n^2+11*n-2)*a(n-1) +(35*n^2-23*n-42)*a(n-2) +(35*n^2-257*n+426)*a(n-3) +3*(-13*n^2+93*n-166)*a(n-4) +2*(n-4)*(2*n-9)*a(n-5)=0. - _R. J. Mathar_, Feb 10 2024
%e G.f.: A(x) = 1 + 2*x + 7*x^2 + 34*x^3 + 195*x^4 + 1225*x^5 +...
%e where g.f. A = A(x) satisfies the equivalent expressions:
%e A = 1 + x*(1-A^4)/(1-A^2) + x^2*(1-A^6)/(1-A^2) + x^3*(1-A^8)/(1-A^2) +...
%e A = 1 + x*(1 + A^2) + x^2*(1 + A^2 + A^4) + x^3*(1 + A^2 + A^4 + A^6) +...
%t Rest[CoefficientList[InverseSeries[Series[(2*x^2)/(1 + x^2 - Sqrt[1 - 4*x - 2*x^2 + x^4]), {x, 0, 30}], x], x]] (* _Vaclav Kotesovec_, Jul 30 2021 *)
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, A^(2*k))+x*O(x^n))); polcoeff(A, n)}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1/((1-x)*(1 - x*A^2+x*O(x^n))));polcoeff(A,n)}
%o (PARI) {a(n)=polcoeff(1/x*serreverse(2*x^2/(1+x^2-sqrt(1-4*x-2*x^2+x^4+x^3*O(x^n)))),n)}
%Y Cf. A007317, A171199, A199548, A349289, A349290, A349291, A349292, A349293.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 08 2011
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