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A323313 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + 1 - (1+x)^n)^(n+1).

%I A323313 #9 Aug 11 2021 17:17:55
%S A323313 1,1,5,42,553,9757,213989,5577285,167819725,5715066723,217100774130,
%T A323313 9097820151376,416870148834739,20734141087348929,1112567525238708108,
%U A323313 64067000804518722936,3941130502347547574782,257950869129924123671281,17899157911139617070534480,1312572773247815201019870104,101429898156047011931583837037,8238300763728888616466106146271,701648737726537575830481016280283
%N A323313 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + 1 - (1+x)^n)^(n+1).
%H A323313 Paul D. Hanna, <a href="/A323313/b323313.txt">Table of n, a(n) for n = 0..300</a>
%F A323313 G.f. A(x) satisfies:
%F A323313 (1) 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + 1 - (1+x)^n)^(n+1).
%F A323313 (2) 1 = Sum_{n>=0} ((1+x)^n + 1)^n/(A(x) + 1 + (1+x)^n)^(n+1).
%F A323313 a(n) ~ c * A317904^n * n^n / exp(n), where c = 0.546536986945124389069... - _Vaclav Kotesovec_, Aug 11 2021
%e A323313 G.f.: A(x) = 1 + x + 5*x^2 + 42*x^3 + 553*x^4 + 9757*x^5 + 213989*x^6 + 5577285*x^7 + 167819725*x^8 + 5715066723*x^9 + 217100774130*x^10 + ...
%e A323313 such that
%e A323313 1 = 1/A(x) + ((1+x) - 1)/(A(x) + 1 - (1+x))^2  +  ((1+x)^2 - 1)^2/(A(x) + 1 - (1+x)^2)^3  +  ((1+x)^3 - 1)^3/(A(x) + 1 - (1+x)^3)^4  +  ((1+x)^4 - 1)^4/(A(x) + 1 - (1+x)^4)^5  +  ((1+x)^5 - 1)^5/(A(x) + 1 - (1+x)^5)^6 + ...
%e A323313 also,
%e A323313 1 = 1/(A(x) + 2)  +  (1 + (1+x))/(A(x) + 1 + (1+x))^2  +  (1 + (1+x)^2)^2/(A(x) + 1 + (1+x)^2)^3  +  (1 + (1+x)^3)^3/(A(x) + 1 + (1+x)^3)^4  +  (1 + (1+x)^4)^4/(A(x) + 1 + (1+x)^4)^5  +  (1 + (1+x)^5)^5/(A(x) + 1 + (1+x)^5)^6 + ...
%o A323313 (PARI) {a(n) = my(A=[1],X=x+x*O(x^n)); for(i=1,n, A=concat(A,0); A[#A] = Vec( sum(m=0, #A, ((1+X)^m - 1)^m / (Ser(A) + 1 - (1+X)^m)^(m+1) ) )[#A]); A[n+1]}
%o A323313 for(n=0,30,print1(a(n),", "))
%Y A323313 Cf. A323311, A323573.
%K A323313 nonn
%O A323313 0,3
%A A323313 _Paul D. Hanna_, Feb 02 2019