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A213639 G.f. A(x) satisfies x = A( x - A(x)^3/x ).

%I A213639 #11 Jun 05 2025 09:55:46
%S A213639 1,1,5,38,357,3832,45189,572378,7676653,107971691,1581714400,
%T A213639 24012849880,376361077578,6071985730614,100602798234000,
%U A213639 1708558136679750,29698002444820760,527661478169200755,9573199146196780335,177192815265794698364,3343432166097650920872
%N A213639 G.f. A(x) satisfies x = A( x - A(x)^3/x ).
%F A213639 G.f. satisfies:
%F A213639 (1) A(x) = x + A(A(x))^3 / A(x).
%F A213639 (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x^n / n!.
%F A213639 (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x^(n+1) / n! ).
%e A213639 G.f.: A(x) = x + x^2 + 5*x^3 + 38*x^4 + 357*x^5 + 3832*x^6 + 45189*x^7 +...
%e A213639 Related series:
%e A213639 A(x)^3/x = x^2 + 3*x^3 + 18*x^4 + 145*x^5 + 1389*x^6 + 14967*x^7 +...
%e A213639 A(A(x)) = x + 2*x^2 + 12*x^3 + 102*x^4 + 1042*x^5 + 11977*x^6 + 149776*x^7 +...
%e A213639 A(A(x))^3/A(x) = x^2 + 5*x^3 + 38*x^4 + 357*x^5 + 3832*x^6 + 45189*x^7 + ...
%e A213639 The g.f. satisfies:
%e A213639 A(x) = x + A(x)^3/x + [d/dx A(x)^6/x^2]/2! + [d^2/dx^2 A(x)^9/x^3]/3! + [d^3/dx^3 A(x)^12/x^4]/4! +...
%e A213639 Logarithmic series:
%e A213639 log(A(x)/x) = A(x)^3/x^2 + [d/dx A(x)^6/x^3]/2! + [d^2/dx^2 A(x)^9/x^4]/3! + [d^3/dx^3 A(x)^12/x^5]/4! +...
%e A213639 From _Seiichi Manyama_, Jun 05 2025: (Start)
%e A213639 Let b(n,k) = [x^n] (A(x)/x)^k.
%e A213639 b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,3*j).
%e A213639 a(n) = b(n-1,1). (End)
%o A213639 (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^3/x+x*O(x^n))); polcoeff(A, n))}
%o A213639 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A213639 {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(3*m)/x^m)/m!)+x*O(x^n)); polcoeff(A, n)}
%o A213639 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A213639 {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(3*m)/x^(m+1))/m!)+x*O(x^n))); polcoeff(A, n)}
%o A213639 for(n=1, 21, print1(a(n), ", "))
%o A213639 (PARI) b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 3*j)));
%o A213639 a(n) = b(n-1, 1); \\ _Seiichi Manyama_, Jun 05 2025
%Y A213639 Cf. A213591.
%K A213639 nonn
%O A213639 1,3
%A A213639 _Paul D. Hanna_, Jun 16 2012