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%I #7 Jun 30 2025 10:17:31
%S 1,1,2,11,64,446,3420,28428,252072,2360784,23187228,237586156,
%T 2529557212,27898101068,317939375512,3736715692256,45216913769794,
%U 562576653920012,7188297232200600,94231521967695334,1266217030228294392,17426887813843435996,245483608643275477496,3536990534237805030068
%N G.f. A(x) satisfies x + x^2 = A(A(x)) - A(A(A(x)))^2.
%H Paul D. Hanna, <a href="/A384831/b384831.txt">Table of n, a(n) for n = 1..500</a>
%e G.f.: A(x) = x + x^2 + 2*x^3 + 11*x^4 + 64*x^5 + 446*x^6 + 3420*x^7 + 28428*x^8 + 252072*x^9 + 2360784*x^10 + 23187228*x^11 + 237586156*x^12 + ...
%e RELATED SERIES.
%e A(A(x)) = x + 2*x^2 + 6*x^3 + 33*x^4 + 216*x^5 + 1612*x^6 + 13204*x^7 + 116272*x^8 + 1086394*x^9 + 10675146*x^10 + ...
%e A(A(A(x))) = x + 3*x^2 + 12*x^3 + 72*x^4 + 518*x^5 + 4184*x^6 + 36776*x^7 + 345365*x^8 + 3424756*x^9 + 35575824*x^10 + ...
%e A(A(A(x)))^2 = x^2 + 6*x^3 + 33*x^4 + 216*x^5 + 1612*x^6 + 13204*x^7 + 116272*x^8 + ...
%o (PARI) {a(n) = my(A, V=[0,1,1]); for(i=1,n, V=concat(V,0); A = Ser(V);
%o V[#V] = polcoef(x + x^2 - subst(A,x,A) + subst(A,x,subst(A,x,A))^2, #V-1)/2 ); V[n+1]}
%o for(n=1,50,print1(a(n),", "))
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jun 29 2025