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%I #16 Feb 24 2018 12:32:56
%S 1,1,2,4,11,35,123,462,1829,7558,32380,143102,649999,3026171,14411412,
%T 70095713,347817785,1759198500,9063638685,47545501777,253854457415,
%U 1379172691108,7623064091313,42860238300826,245098499411379,1425403070154478,8429327482836740,50681175605982771
%N G.f. satisfies: A(x) = Sum_{n>=0} x^n*(1 + x*A(x)^n)^n.
%C Compare g.f. to a g.f. C(x) of the Catalan sequence:
%C C(x) = Sum_{n>=0} x^n*(1 + x*C(x)^2)^n where C(x) = 1 + x*C(x)^2.
%H Paul D. Hanna, <a href="/A186998/b186998.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. satisfies: A(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2) / (1 - x*A(x)^n)^(n+1). - _Paul D. Hanna_, Sep 24 2014
%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 35*x^5 + 123*x^6 +...
%e such that
%e A(x) = 1 + x*(1+x*A(x)) + x^2*(1+x*A(x)^2)^2 + x^3*(1+x*A(x)^3)^3 + x^4*(1+x*A(x)^4)^4 + x^5*(1+x*A(x)^5)^5 + x^6*(1+x*A(x)^6)^6 +...
%e The g.f. satisfies the series identity:
%e A(x) = 1/(1-x) + x^2*A(x)/(1-x*A(x))^2 + x^4*A(x)^4/(1-x*A(x)^2)^3 + x^6*A(x)^9/(1-x*A(x)^3)^4 + x^8*A(x)^16/(1-x*A(x)^4)^5 + x^10*A(x)^25/(1-x*A(x)^5)^6 +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(1+x*(A+x*O(x^n))^m)^m));polcoeff(A,n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n,x^(2*k)*A^(k^2)/(1 - x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Sep 24 2014
%Y Cf. A186999, A203000, A247330, A186999.
%Y Cf. A300041, A300042, A300043.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 01 2011
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