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A143435 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^3.

%I A143435 #9 Jun 05 2025 09:52:33
%S A143435 1,1,3,15,97,738,6297,58630,585543,6200916,69071103,804470751,
%T A143435 9753459717,122670681073,1596129692136,21437840848440,296680980737270,
%U A143435 4224090724829151,61794432127467450,927795254532531834,14282871462981487854,225247807261125989496,3636185180695164503129
%N A143435 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^3.
%F A143435 G.f. satisfies: x - G(x) = G(x)^2*A(x)^3 where G(x*A) = x.
%F A143435 G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(3*n+3)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]
%F A143435 From _Seiichi Manyama_, Jun 05 2025: (Start)
%F A143435 Let a(n,k) = [x^n] A(x)^k.
%F A143435 a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(n-j+k,j)/(n-j+k) * a(n-j,3*j). (End)
%e A143435 G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 97*x^4 + 738*x^5 + 6297*x^6 +...
%e A143435 A(x*A(x)) = 1 + x + 4*x^2 + 24*x^3 + 178*x^4 + 1511*x^5 + 14130*x^6 +...
%e A143435 A(x*A(x))^3 = 1 + 3*x + 15*x^2 + 97*x^3 + 738*x^4 + 6297*x^5 +...
%e A143435 Logarithmic series:
%e A143435 log(A(x)) = x*A(x) + [d/dx x^3*A(x)^6]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^9]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^12]*A(x)^(-8)/4! +...
%o A143435 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^3,x,x*A));polcoeff(A,n)}
%o A143435 (PARI) /* n-th Derivative: */
%o A143435 {Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D}
%o A143435 /* G.f.:  [Paul D. Hanna, Dec 18 2010] */
%o A143435 {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,
%o A143435 A=exp(sum(m=0,n,Dx(m,x^(2*m+1)*A^(3*m+3))*A^(-2*m-2)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}
%o A143435 (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, 3*j))); \\ _Seiichi Manyama_, Jun 05 2025
%Y A143435 Cf. A143426, A143436, A143437.
%Y A143435 Cf. A139702, A087949, A182969.
%K A143435 nonn
%O A143435 0,3
%A A143435 _Paul D. Hanna_, Aug 14 2008