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A143426 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^2.

%I A143426 #13 Jun 05 2025 09:52:29
%S A143426 1,1,2,7,32,175,1086,7429,54994,435120,3647686,32192596,297654824,
%T A143426 2872372828,28841766844,300592170551,3244942353856,36219458512421,
%U A143426 417365572999944,4958429472475171,60659660219655616,763325035692109389,9870492111677035538
%N A143426 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^2.
%H A143426 Vaclav Kotesovec, <a href="/A143426/b143426.txt">Table of n, a(n) for n = 0..235</a>
%F A143426 G.f. satisfies: x - G(x) = G(x)^2*A(x)^2 where G(x*A(x)) = x.
%F A143426 G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(2n+2)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
%F A143426 From _Seiichi Manyama_, Jun 05 2025: (Start)
%F A143426 Let a(n,k) = [x^n] A(x)^k.
%F A143426 a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(n-j+k,j)/(n-j+k) * a(n-j,2*j). (End)
%e A143426 G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 175*x^5 + 1086*x^6 +...
%e A143426 A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 434*x^5 + 2986*x^6 +...
%e A143426 A(x*A(x))^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 175*x^4 + 1086*x^5 +...
%e A143426 Logarithmic series:
%e A143426 log(A(x)) = x + [d/dx x^3*A(x)^4]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^6]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^8]*A(x)^(-8)/4! +...
%o A143426 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^2,x,x*A));polcoeff(A,n)}
%o A143426 (PARI) /* n-th Derivative: */
%o A143426 {Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D}
%o A143426 /* G.f.: [Paul D. Hanna, Dec 18 2010] */
%o A143426 {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,
%o A143426 A=exp(sum(m=0,n,Dx(m,x^(2*m+1)*A^(2*m+2))*A^(-2*m-2)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}
%o A143426 (PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 2*j))); \\ _Seiichi Manyama_, Jun 05 2025
%Y A143426 Cf. A139702, A087949, A143435, A182969.
%Y A143426 Cf. A143500.
%K A143426 nonn
%O A143426 0,3
%A A143426 _Paul D. Hanna_, Aug 14 2008