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A276365 G.f. A(x) satisfies: A(x - 2*A(x)^2) = x - A(x)^2.

%I A276365 #18 Nov 18 2022 20:06:10
%S A276365 1,1,6,53,578,7234,100044,1495125,23802346,399740086,7032766196,
%T A276365 128952474242,2454645604820,48359400068836,983683769369624,
%U A276365 20618782389897333,444636132851851386,9851377271964349038,223998085060636514020,5221799494107885481430,124695762315403816775932,3047952254964607540099676,76206565881709345978097960,1947752912315470845518308642,50860833685759573411702643972
%N A276365 G.f. A(x) satisfies: A(x - 2*A(x)^2) = x - A(x)^2.
%H A276365 Paul D. Hanna, <a href="/A276365/b276365.txt">Table of n, a(n) for n = 1..300</a>
%F A276365 G.f. A(x) also satisfies:
%F A276365 (1) A(x) = x + A( 2*A(x) - x )^2.
%F A276365 (2) 2*A(x) = x + Series_Reversion(x - 2*A(x)^2).
%F A276365 (3) R(x) = 2*x - Series_Reversion(x - A(x)^2), where R(A(x)) = x.
%F A276365 (4) R( (x - R(x))^(1/2) ) = 2*x - R(x), where R(A(x)) = x.
%F A276365 (5) A(x) = x + Sum_{n>=1} 2^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
%F A276365 (6) A(x) = x + G(A(x))^2, where G(x) = sqrt(x - R(x)) is the g.f. of A177409, and R(A(x)) = x. - _Paul D. Hanna_, Nov 18 2022
%F A276365 a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k.
%e A276365 G.f.: A(x) = x + x^2 + 6*x^3 + 53*x^4 + 578*x^5 + 7234*x^6 + 100044*x^7 + 1495125*x^8 + 23802346*x^9 + 399740086*x^10 + 7032766196*x^11 +...
%e A276365 such that A(x - 2*A(x)^2) = x - A(x)^2.
%e A276365 RELATED SERIES.
%e A276365 Note that Series_Reversion(x - 2*A(x)^2) = 2*A(x) - x, which begins:
%e A276365 Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1156*x^5 + 14468*x^6 + 200088*x^7 + 2990250*x^8 + 47604692*x^9 + 799480172*x^10 +...
%e A276365 Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
%e A276365 R(x) = x - x^2 - 4*x^3 - 28*x^4 - 264*x^5 - 2992*x^6 - 38496*x^7 - 544464*x^8 - 8298080*x^9 - 134500672*x^10 - 2297361024*x^11 +...
%e A276365 then Series_Reversion(x - A(x)^2) = 2*x - R(x), and
%e A276365 R(x) = x - G(x)^2, where G(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1208*x^5 + 15536*x^6 + 220832*x^7 + 3390480*x^8 + ... + A177409(n)*x^n + ...
%e A276365 Also, sqrt(A(x) - x) = A(2*A(x) - x), which begins:
%e A276365 sqrt(A(x) - x) = x + 3*x^2 + 22*x^3 + 223*x^4 + 2706*x^5 + 36998*x^6 + 552172*x^7 + 8827263*x^8 + 149328698*x^9 + 2650946274*x^10 + ...
%t A276365 m = 26; A[_] = 0;
%t A276365 Do[A[x_] = x + A[2 A[x] - x]^2 + O[x]^m // Normal, {m}];
%t A276365 CoefficientList[A[x]/x, x] (* _Jean-François Alcover_, Sep 30 2019 *)
%o A276365 (PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 2*F^2) + F^2, #A) ); A[n]}
%o A276365 for(n=1, 30, print1(a(n), ", "))
%Y A276365 Cf. A277295, A213591, A275765, A177409.
%K A276365 nonn
%O A276365 1,3
%A A276365 _Paul D. Hanna_, Sep 01 2016