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A143500 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2).

%I A143500 #11 Jun 04 2025 09:52:43
%S A143500 1,1,1,3,10,46,244,1481,10020,74400,599573,5200284,48223360,475557054,
%T A143500 4965035754,54672110310,632853655686,7678552433184,97404631390960,
%U A143500 1288861146261679,17752479062092470,254051633672160696,3770953046565933003,57964955567444706668
%N A143500 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2).
%H A143500 Vaclav Kotesovec, <a href="/A143500/b143500.txt">Table of n, a(n) for n = 0..230</a>
%F A143500 G.f. satisfies: G(x) = x/[1 + A(x)*G(x)]^2 = x/A(G(x))^2 where G(x*A(x)^2) = x.
%F A143500 From _Seiichi Manyama_, Jun 04 2025: (Start)
%F A143500 Let a(n,k) = [x^n] A(x)^k.
%F A143500 a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+k,j)/(2*n-2*j+k) * a(n-j,j). (End)
%e A143500 G.f. A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 46*x^5 + 244*x^6 +...
%e A143500 A(x)^2 = 1 + 2*x + 3*x^2 + 8*x^3 + 27*x^4 + 118*x^5 + 609*x^6 +...
%e A143500 A(x*A(x)^2) = 1 + x + 3*x^2 + 10*x^3 + 46*x^4 + 244*x^5 +...
%e A143500 If G(x*A(x)^2) = x then
%e A143500 G(x) = x - 2*x^2 + 5*x^3 - 18*x^4 + 68*x^5 - 300*x^6 + 1283*x^7 -+...
%e A143500 A(G(x)) = 1 + A(x)*G(x) = (x/G(x))^(1/2) where
%e A143500 A(x)*G(x) = x - x^2 + 4*x^3 - 12*x^4 + 59*x^5 - 209*x^6 + 1199*x^7 -...
%o A143500 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,x*A^2));polcoeff(A,n)}
%o A143500 (PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, j))); \\ _Seiichi Manyama_, Jun 04 2025
%Y A143500 Cf. A087949, A143501.
%K A143500 nonn
%O A143500 0,4
%A A143500 _Paul D. Hanna_, Aug 20 2008