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a(n) ~ c * 2^n * (n-1)!, where c = 0.1261880758068409567445... - Vaclav Kotesovec, Oct 20 2020
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More generally, [x^n] G(x,k)^(k*(n+1)-1) / (x*G(x,k)^k)' = 0 is satisfied by an integer sequence series G(x,k) when k is a fixed positive integer.
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G.f. A(x) satisfies: [x^n] A(x)^n / (x*A(x))' = 0 for n > 1.
Odd terms seem to occur only at positions 0, 1, and 2*A118113(k) for k >= 0.
More generally, [x^n] G(x,k)^(k*(n+1)-1) / (x*G(x,k)^k)' = 0 is satisfied by an integer series sequence G(x,k) when k is a fixed positive integer.
G.f. A(x) satisfies: [x^n] A(x)^n / (A(x) + x*A'(x)) = 0 for n > 1.
such that [x^n] A(x)^n / (x*A(x))' = 0 for n > 1.
in which the main diagonal consists of all zeros after the initial terms, illustrating that [x^n] A(x)^n / (x*A(x))' = 0 for n > 1.
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Compare to identity: [x^n] (x*F(x))' / F(x)^(n+1) = 0 holds when F(0) = 1.
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