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%I #24 Mar 03 2024 13:46:49
%S 1,1,3,14,78,477,3094,20905,145547,1036891,7522335,55382045,412740298,
%T 3107671807,23604165913,180641336755,1391555475647,10781886600707,
%U 83968131035849,656931982467460,5160714860765430,40692065290732340,321937030883130021
%N G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x)^4).
%F a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+2*k,n-1-k) for n > 0.
%F From _Paul D. Hanna_, Aug 19 2023: (Start)
%F A(x)^m = Sum_{n>=0} a(n,m) * x^n, where a(n,m) = (m/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+2*k+m-1,n-1-k) for n > 0.
%F A(x) = exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{k=0..n-1} binomial(n,k) * binomial(2*n+2*k-1,n-1-k) for n >= 1.
%F G.f. A(x) satisfies the following formulas.
%F (1) A(x) = 1 + x*(A(x)^2 - A(x)^4 + A(x)^5).
%F (2) A(x)^2 = 1 + x*(A(x)^2 + A(x)^3 - A(x)^4 + A(x)^6).
%F (3) A(x)^3 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^7).
%F (4) A(x)^4 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^5 + A(x)^8).
%F (5) A(x)^5 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^5 + A(x)^6 + A(x)^9).
%F (6) A(x)^6 = 1 + x*(A(x)^2 + A(x)^3 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^10).
%F (End)
%t nmax = 22; A[_] = 1;
%t Do[A[x_] = 1 + x*A[x]^2/(1-x*A[x]^4) + O[x]^(nmax+1) // Normal, {nmax+1}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Mar 03 2024 *)
%o (PARI) a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(2*n+2*k, n-1-k))/n);
%Y Cf. A001003, A001764, A219537, A364740.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 05 2023
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