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A323573 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + (1+x)^n - 1)^(n+1).

%I A323573 #6 Aug 11 2021 17:35:50
%S A323573 1,1,1,6,41,381,4387,59961,946119,16901219,336924252,7412401676,
%T A323573 178372705409,4660680881897,131410732869312,3977081948965664,
%U A323573 128600945014475040,4424941538152614645,161433547224627797940,6224586371820817112652,252934418382142622780667,10803348636926511625239387,483881915960470248201012949
%N A323573 G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^n - 1)^n/(A(x) + (1+x)^n - 1)^(n+1).
%H A323573 Paul D. Hanna, <a href="/A323573/b323573.txt">Table of n, a(n) for n = 0..200</a>
%F A323573 a(n) ~ c * n^n / (exp(n) * log(2)^(2*n)), where c = 0.51205951699411... - _Vaclav Kotesovec_, Aug 11 2021
%e A323573 G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 41*x^4 + 381*x^5 + 4387*x^6 + 59961*x^7 + 946119*x^8 + 16901219*x^9 + 336924252*x^10 + 7412401676*x^11 + 178372705409*x^12 + ...
%e A323573 such that
%e A323573 1 = 1/A(x) +  x/(A(x) + x)^2 + ((1+x)^2 - 1)^2/(A(x) + (1+x)^2 - 1)^3 + ((1+x)^3 - 1)^3/(A(x) + (1+x)^3 - 1)^4 + ((1+x)^4 - 1)^4/(A(x) + (1+x)^4 - 1)^5 + ((1+x)^5 - 1)^5/(A(x) + (1+x)^5 - 1)^6 + ((1+x)^2 - 1)^6/(A(x) + (1+x)^6 - 1)^7 + ...
%o A323573 (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 A323573 for(n=0, 30, print1(a(n), ", "))
%Y A323573 Cf. A323313.
%K A323573 nonn
%O A323573 0,4
%A A323573 _Paul D. Hanna_, Feb 07 2019