A301434 - OEIS

%I #3 Mar 25 2018 16:28:02

%S 1,1,5,44,515,7475,129525,2619384,60823498,1600232884,47150091652,

%T 1539874276303,55237696912990,2159149195744518,91335745732513695,

%U 4156560037257263852,202458583588324775958,10507835578532315131098,578860329653382364799458,33730837645131194789676597,2072763842696435360327368451,133954882943472707055024904486,9082145193772600621881071937985

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

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

%e G.f.: A(x) = 1 + x + 5*x^2 + 44*x^3 + 515*x^4 + 7475*x^5 + 129525*x^6 + 2619384*x^7 + 60823498*x^8 + 1600232884*x^9 + 47150091652*x^10 + ...

%e such that

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

%e Also,

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

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

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

%Y Cf. A301433.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 25 2018