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A275765 G.f. satisfies: A(x - A(x)^2) = x + A(x)^2.

%I A275765 #23 Sep 30 2019 02:47:40
%S A275765 1,2,12,106,1148,14156,191400,2775930,42585412,684496988,11449962008,
%T A275765 198331811356,3543990791480,65136985937096,1228531761076208,
%U A275765 23733123786608826,468887742020767788,9461919438245032500,194817077269127033944,4089069139317823277548,87426000975842460304792,1902787414323673070857528,42133267254272433484761584,948695717599714654940068604,21712101305047777916075831096,504865916349551192319293625016
%N A275765 G.f. satisfies: A(x - A(x)^2) = x + A(x)^2.
%H A275765 Paul D. Hanna, <a href="/A275765/b275765.txt">Table of n, a(n) for n = 1..300</a>
%F A275765 G.f. A(x) also satisfies:
%F A275765 (1) A(x) = x + 2 * A( x/2 + A(x)/2 )^2.
%F A275765 (2) A(x) = -x + 2 * Series_Reversion(x - A(x)^2).
%F A275765 (3) R(x) = -x + 2 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
%F A275765 (4) R( sqrt( x/2 - R(x)/2 ) ) = x/2 + R(x)/2, where R(A(x)) = x.
%F A275765 a(n) = Sum_{k=0..n-1} A277295(n,k)*2^(n-k).
%e A275765 G.f.: A(x) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1148*x^5 + 14156*x^6 + 191400*x^7 + 2775930*x^8 + 42585412*x^9 + 684496988*x^10 + 11449962008*x^11 + 198331811356*x^12 +...
%e A275765 such that A(x - A(x)^2) = x + A(x)^2.
%e A275765 RELATED SERIES.
%e A275765 Series_Reversion(x - A(x)^2) = x + x^2 + 6*x^3 + 53*x^4 + 574*x^5 + 7078*x^6 + 95700*x^7 + 1387965*x^8 + 21292706*x^9 + 342248494*x^10 +...
%e A275765 which equals (A(x) + x)/2.
%e A275765 A( (A(x) + x)/2 ) = x + 3*x^2 + 22*x^3 + 221*x^4 + 2634*x^5 + 35086*x^6 + 506356*x^7 + 7773279*x^8 + 125441594*x^9 + 2110832382*x^10 +...
%e A275765 which equals sqrt( (A(x) - x)/2 ).
%e A275765 Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
%e A275765 R(x) = x - 2*x^2 - 4*x^3 - 26*x^4 - 228*x^5 - 2396*x^6 - 28440*x^7 - 369114*x^8 - 5135468*x^9 - 75602108*x^10 - 1167066216*x^11 - 18768202924*x^12 +...
%e A275765 then Series_Reversion(x + A(x)^2) = x/2 + R(x)/2.
%t A275765 m = 26; A[_] = 0;
%t A275765 Do[A[x_] = x + 2 A[x/2 + A[x]/2]^2 + O[x]^(m+1) // Normal, {m+1}];
%t A275765 CoefficientList[A[x]/x, x] (* _Jean-François Alcover_, Sep 30 2019 *)
%o A275765 (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-F^2) - F^2,#A) );A[n]}
%o A275765 for(n=1,30,print1(a(n),", "))
%Y A275765 Cf. A277295, A213591, A276364, A276360, A276361, A276362, A276363.
%K A275765 nonn
%O A275765 1,2
%A A275765 _Paul D. Hanna_, Aug 31 2016