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%I #8 Jun 05 2019 03:27:33
%S 1,1,2,20,282,5134,112053,2823119,80202565,2529045393,87523776013,
%T 3295995672161,134155142687732,5869278171065418,274718037952537674,
%U 13701118397652347442,725505704889894172448,40658992718689480518864,2404662897766073643050293,149692182669205551972626617,9784886698908632846522031701
%N G.f. A(x) satisfies: 0 = Sum_{n>=1} n * ((1+x)^n - A(x))^n.
%H Paul D. Hanna, <a href="/A326010/b326010.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) 0 = Sum_{n>=1} n * ((1+x)^n - A(x))^n.
%F (2) A(x) = P(x)/Q(x) where
%F P(x) = Sum_{n>=0} n * (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+2),
%F Q(x) = Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n*A(x))^(n+2).
%F (3) A'(x) = P(x)/Q(x) where
%F P(x) = Sum_{n>=0} (n+1)^3 * ((1+x)^(n+1) - A(x))^n * (1+x)^n,
%F Q(x) = Sum_{n>=0} (n+1)^2 * ((1+x)^(n+1) - A(x))^n.
%F a(n) ~ c * d^n * sqrt(n) * n!, where d = A317855 = 3.16108865386... and c = 0.102568345138... - _Vaclav Kotesovec_, Jun 05 2019
%e G.f.: A(x) = 1 + x + 2*x^2 + 20*x^3 + 282*x^4 + 5134*x^5 + 112053*x^6 + 2823119*x^7 + 80202565*x^8 + 2529045393*x^9 + 87523776013*x^10 + ...
%e such that
%e 0 = ((1+x) - A(x)) + 2*((1+x)^2 - A(x))^2 + 3*((1+x)^3 - A(x))^3 + 4*((1+x)^4 - A(x))^4 + 5*((1+x)^5 - A(x))^5 + 6*((1+x)^6 - A(x))^6 + ...
%e The terms a(n) modulo 2 begin:
%e 1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,
%e 0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,0,0,
%e 0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0,
%e 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,1,1,
%e 0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,1,
%e 1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,1,
%e 0,0,0,0,1,1,1,1,0,0,1,1,1,1,1,1,0,0,1,1,1,1,0,0,
%e 1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,
%e 1,1,0,0,0,0,0,0,0, ...
%o (PARI) {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = polcoeff( sum(m=1,#A, m* ((1+x)^m - Ser(A))^m ), #A-1));A[n+1]}
%o for(n=0,25,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 04 2019
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