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A374567 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).

%I A374567 #12 Aug 13 2024 22:54:44
%S A374567 1,2,9,51,325,2222,15926,118085,898217,6970053,54960439,439112322,
%T A374567 3547096393,28921270773,237704587991,1967321998468,16381661824340,
%U A374567 137144132047520,1153655788549216,9746264972136632,82656795697147384,703459159019830315,6005956718852682504,51426768620398474939
%N A374567 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).
%C A374567 Compare to: C(x)^2 = C( x^2 - 2*C(x)^3 ), where C(x) = x - C(x)^2.
%H A374567 Paul D. Hanna, <a href="/A374567/b374567.txt">Table of n, a(n) for n = 1..500</a>
%F A374567 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A374567 (1) A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).
%F A374567 (2) x = A( x - x^2 - x*G(x) ), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of A001190, the Wedderburn-Etherington numbers.
%F A374567 (3) x^2 = A( x^2*(1 - G(x))^2 + 2*x^3 - x^4 ), where G(x) is the g.f. of A001190.
%F A374567 (4) x = A( x*sqrt(1 - 2*x - G(x^2)) - x^2 ), where G(x) is the g.f. of A001190.
%e A374567 G.f.: A(x) = x + 2*x^2 + 9*x^3 + 51*x^4 + 325*x^5 + 2222*x^6 + 15926*x^7 + 118085*x^8 + 898217*x^9 + 6970053*x^10 + ...
%e A374567 where A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).
%e A374567 RELATED SERIES.
%e A374567 Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then
%e A374567 A( x - x^2 - x*G(x) ) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) begins
%e A374567 G(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + 207*x^11 + 451*x^12 + 983*x^13 + ... + A001190(n)*x^n + ...
%e A374567 A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 138*x^5 + 935*x^6 + 6662*x^7 + 49191*x^8 + 373020*x^9 + 2887711*x^10 + 22727256*x^11 + ...
%e A374567 A(x)^3 = x^3 + 6*x^4 + 39*x^5 + 269*x^6 + 1938*x^7 + 14418*x^8 + 109932*x^9 + 854568*x^10 + 6747672*x^11 + ...
%e A374567 SPECIFIC VALUES.
%e A374567 A(t) = 1/5 at t = 0.1094430388151747748055350980742058560407673560783455...
%e A374567 where 1/25 = A( t^2 + 2*t/25 + 2/125 ).
%e A374567 A(t) = 1/6 at t = 0.1053569291935061227625330002451062383852684202941979...
%e A374567 where 1/36 = A( t^2 + t/18 + 1/108 ).
%e A374567 A(1/10) = 0.1471263013840628871589336795118257882025452972700045...
%e A374567 where A(1/10)^2 = A( 1/10^2 + (2/10)*A(1/10)^2 + 2*A(1/10)^3 ).
%e A374567 A(1/11) = 0.1237258078822115859596611191115342221543387518134407...
%e A374567 A(1/12) = 0.1081759735424269717469930892718654709953905803313352...
%o A374567 (PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
%o A374567 A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2 + 2*x*Ax^2 + 2*Ax^3) - Ax^2, #A) ); A[n+1]}
%o A374567 for(n=1, 30, print1(a(n), ", "))
%Y A374567 Cf. A374566, A271959, A001190.
%K A374567 nonn
%O A374567 1,2
%A A374567 _Paul D. Hanna_, Aug 13 2024