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A387041 G.f. A(x) satisfies (A(x) - x^2) o (x - A(x)^2) = x, where operator 'o' denotes composition.

%I A387041 #12 Aug 15 2025 10:28:49
%S A387041 1,2,6,41,348,3360,35632,406104,4904914,62180918,821752456,
%T A387041 11263836924,159523476148,2327336091732,34894961587312,
%U A387041 536671299862721,8453184479505430,136188177741639378,2241801065131393700,37670062720274627960,645649822816127973456,11279877783091509190416
%N A387041 G.f. A(x) satisfies (A(x) - x^2) o (x - A(x)^2) = x, where operator 'o' denotes composition.
%H A387041 Paul D. Hanna, <a href="/A387041/b387041.txt">Table of n, a(n) for n = 1..520</a>
%F A387041 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A387041 (1) A(x) = x + x^2 + A(A(x) - x^2)^2.
%F A387041 (2) A(x - A(x)^2) = x + (x - A(x)^2)^2.
%F A387041 (3) A(A(x) - x^2) = sqrt( A(x) - x^2 - x ).
%F A387041 (4) A(x)^2 = x - sqrt( A(x - A(x)^2) - x ).
%F A387041 (5) A(x) = x^2 + Series_Reversion(x - A(x)^2).
%F A387041 (6) A(x) = sqrt( x - Series_Reversion(A(x) - x^2) ).
%F A387041 (7) A(x) = x + x^2 + Sum_{n>=0} d^n/dx^n A(x)^(2*n+2) / (n+1)!.
%F A387041 (8) A(x) = x^2 + x*exp( Sum_{n>=0} d^n/dx^n (A(x)^(2*n+2)/x) / (n+1)! ).
%e A387041 G.f.: A(x) = x + 2*x^2 + 6*x^3 + 41*x^4 + 348*x^5 + 3360*x^6 + 35632*x^7 + 406104*x^8 + 4904914*x^9 + 62180918*x^10 + ...
%e A387041 where A(x) - x^2 = x + A(A(x) - x^2)^2;
%e A387041 also, A(x - A(x)^2) = x + (x - A(x)^2)^2 = x + x^2 - 2*x*A(x)^2 + A(x)^4.
%e A387041 RELATED SERIES.
%e A387041 A(A(x) - x^2) = x + 3*x^2 + 16*x^3 + 126*x^4 + 1174*x^5 + 12278*x^6 + 139496*x^7 + 1689597*x^8 + 21553566*x^9 + 287191110*x^10 + ...
%e A387041 A(x - A(x)^2) = x + x^2 - 2*x^3 - 7*x^4 - 24*x^5 - 164*x^6 - 1452*x^7 - 14312*x^8 - 153354*x^9 - 1757322*x^10 - ...
%e A387041 A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 106*x^5 + 896*x^6 + 8604*x^7 + 90561*x^8 + 1023592*x^9 + 12258452*x^10 + ...
%e A387041 A(x)^4 = x^4 + 8*x^5 + 48*x^6 + 340*x^7 + 2896*x^8 + 27768*x^9 + 289862*x^10 + ...
%o A387041 (PARI) {a(n) = my(V=[0,1]); for(i=1,n, V = concat(V,0); A = Ser(V);
%o A387041 V[#V] = polcoef(x + (x - A^2)^2 - subst(A,x, x - A^2),#V-1)); polcoef(A,n)}
%o A387041 for(n=1,30,print1(a(n),", "))
%K A387041 nonn
%O A387041 1,2
%A A387041 _Paul D. Hanna_, Aug 14 2025