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%I #8 Jun 28 2024 05:06:14
%S 1,2,10,66,498,4056,34644,305310,2749110,25142172,232728588,
%T 2176116348,20532197196,195344525540,1872680305544,18073069864926,
%U 175419949070118,1710976713480396,16761489153049788,164888041322062428,1628416166697339324,16136415431311552992,160333972547949898584
%N Expansion of g.f. A(x) satisfying A( A(x) - 4*A(x)^2 + 4*A(x)^3 ) = x.
%C a(38) = -19599187894639322176080463718044944 is the first negative term.
%H Paul D. Hanna, <a href="/A372580/b372580.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F (1) x = A( A(x)*(1 - 2*A(x))^2 ).
%F (2) x = A(A( x*(1 - 2*x)^2 )).
%F (3) x = B(x)*(1 - 2*B(x))^2 where B(x) = A(A(x)) (A369510).
%F (4) A(x) = Series_Reversion( A(x)*(1 - 2*A(x))^2 ).
%e G.f.: A(x) = x + 2*x^2 + 10*x^3 + 66*x^4 + 498*x^5 + 4056*x^6 + 34644*x^7 + 305310*x^8 + 2749110*x^9 + 25142172*x^10 + 232728588*x^11 + 2176116348*x^12 + ...
%e where A( A(x)*(1 - 2*A(x))^2 ) = x.
%e RELATED SERIES.
%e Let B(x) = A(A(x)), then B(x)/x is the g.f. of A369510:
%e B(x) = x + 4*x^2 + 28*x^3 + 240*x^4 + 2288*x^5 + 23296*x^6 + 248064*x^7 + 2728704*x^8 + 30764800*x^9 + ... + 2^(n-1)*binomial(3*n-2,n-1)/n * x^n + ...
%e where B(x)*(1 - 2*B(x))^2 = x.
%e Let R(x) be the series reversion, R(A(x)) = x, then
%e R(x) = A(x)*(1 - 2*A(x))^2 = x - 2*x^2 - 2*x^3 - 6*x^4 - 22*x^5 - 80*x^6 - 228*x^7 - 18*x^8 + 6694*x^9 + ... + (-1)^(n-1)*A097090(n)*x^n + ...
%e where R(R(x)) = x*(1 - 2*x)^2.
%o (PARI) {a(n) = my(A=x); for(k=2,n+1, A=truncate(A); A = (A + serreverse( A*(1 - 2*A)^2 +x*O(x^k)))/2 ); polcoeff(A,n)}
%o for(n=1,30, print1(a(n),", "))
%Y Cf. A097090, A369510, A006013.
%K sign
%O 1,2
%A _Paul D. Hanna_, Jun 27 2024
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