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G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (1 + (1+x)^n * A(x))^n.
2

%I #5 Apr 07 2018 15:55:17

%S 1,2,7,33,180,1071,6776,44907,308903,2192218,15986187,119458754,

%T 913085921,7130596660,56856959415,462789536503,3845600755712,

%U 32634320897898,282973974102022,2508805646347474,22758825619762568,211400392340768128,2011934295358876840,19628847839908163287,196377194100036473222,2014897715206819032244,21199978342486061320347,228666292563823622317191

%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (1 + (1+x)^n * A(x))^n.

%F G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * (1+x)^(n^2) / (1 - x*(1+x)^n)^(n+1).

%e G.f.: A(x) = 1 + 2*x + 7*x^2 + 33*x^3 + 180*x^4 + 1071*x^5 + 6776*x^6 + 44907*x^7 + 308903*x^8 + 2192218*x^9 + 15986187*x^10 + ...

%e such that

%e A(x) = 1 + x*(1 + (1+x)*A(x)) + x^2*(1 + (1+x)^2*A(x))^2 + x^3*(1 + (1+x)^3*A(x))^3 + x^4*(1 + (1+x)^4*A(x))^4 + x^5*(1 + (1+x)^5*A(x))^5 + x^6*(1 + (1+x)^6*A(x))^6 + ...

%e Also,

%e A(x) = 1/(1-x) + x*A(x)*(1+x)/(1 - x*(1+x))^2 + x^2*A(x)^2*(1+x)^4/(1 - x*(1+x)^2)^3 + x^3*A(x)^3*(1+x)^9/(1 - x*(1+x)^3)^4 + x^4*A(x)^4*(1+x)^16/(1 - x*(1+x)^4)^5 + ...

%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, x^m * (1 + A*(1+x +x*O(x^n))^m)^m ) );polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A302061, A301434.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 25 2018

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