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A380678 G.f. A(x) satisfies A( x - A(x)^2/(1 - A(x)^2) ) = x.

%I A380678 #9 Feb 15 2025 09:47:39
%S A380678 1,1,4,25,190,1645,15652,160186,1739032,19838179,236192158,2920269202,
%T A380678 37352521348,492799406899,6690428699026,93293086422514,
%U A380678 1334088426585850,19538994465481000,292775222237716612,4484180296611470218,70146488080451823382,1119964903188050808163,18239593214541431577550
%N A380678 G.f. A(x) satisfies A( x - A(x)^2/(1 - A(x)^2) ) = x.
%C A380678 Conjecture: for n>= 1, a(n) is odd iff n = A027383(k) for k >= 0, where A027383(2*m) = 3*2^m - 2 and A027383(2*m+1) = 4*2^m - 2.
%H A380678 Paul D. Hanna, <a href="/A380678/b380678.txt">Table of n, a(n) for n = 1..520</a>
%F A380678 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A380678 (1) A( x - A(x)^2/(1 - A(x)^2) ) = x.
%F A380678 (2) A(x) = x + A(A(x))^2/(1 - A(A(x))^2).
%F A380678 (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)/(1 - A(x)^2)^n / n!.
%F A380678 (4) A(x) = x + B(A(x)) and A(x - B(x)) = x where B(x) = A(x)^2/(1 - A(x)^2) is the g.f. of A380558.
%e A380678 G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1645*x^6 + 15652*x^7 + 160186*x^8 + 1739032*x^9 + 19838179*x^10 + ...
%e A380678 where A(x - A(x)^2/(1 - A(x)^2)) = x.
%e A380678 Let B(x) = x - Series_Reversion(A(x)), where
%e A380678 B(x) = x^2 + 2*x^3 + 10*x^4 + 62*x^5 + 469*x^6 + 4028*x^7 + 37984*x^8 + 385202*x^9 + 4144798*x^10 + ... + A380558(n)*x^n + ...
%e A380678 then A(x) = x + B(A(x)) and B(x - B(x)) = x^2/(1 - x^2).
%o A380678 (PARI) /* Generates N terms of this sequence */
%o A380678 N = 40; A=x; for(m=1,N, A=truncate(A); A = serreverse(x - A^2/(1 - A^2 +O(x^m)) ); ); Vec(A)
%Y A380678 Cf. A380558, A213591, A027383.
%K A380678 nonn
%O A380678 1,3
%A A380678 _Paul D. Hanna_, Feb 14 2025