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A259606 G.f. satisfies: A(x) = Series_Reversion( x - A'(x)*A(x)^2 ).

%I A259606 #22 Oct 13 2024 13:24:21
%S A259606 1,1,6,60,790,12488,226176,4567245,101057170,2421311002,62292579316,
%T A259606 1709994461396,49844902545256,1536870296603860,49965056185462360,
%U A259606 1708221871912841430,61272046476315041664,2301058164207089144028,90309756129843950212480,3697832634432220792202296
%N A259606 G.f. satisfies: A(x) = Series_Reversion( x - A'(x)*A(x)^2 ).
%H A259606 Vaclav Kotesovec, <a href="/A259606/b259606.txt">Table of n, a(n) for n = 1..300</a>
%F A259606 G.f. satisfies:
%F A259606 (1) A(x) = x + A'(A(x)) * A(A(x))^2.
%F A259606 (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A'(x)^n * A(x)^(2*n) / n!.
%F A259606 (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A'(x)^n * A(x)^(2*n) / (n!*x) ).
%F A259606 a(n) ~ c * n! * n^(alfa) / LambertW(1)^n, where alfa = 2.750027682144251700567... and c = 0.005275216890926771261... - _Vaclav Kotesovec_, Aug 25 2017
%F A259606 alfa = 5*LambertW(1) - 2 + 3/(1 + LambertW(1)). - _Vaclav Kotesovec_, Mar 13 2023
%e A259606 G.f.: A(x) = x + x^2 + 6*x^3 + 60*x^4 + 790*x^5 + 12488*x^6 + 226176*x^7 + 4567245*x^8 + 101057170*x^9 + 2421311002*x^10 + ...
%e A259606 where
%e A259606 A(x - A'(x)*A(x)^2) = x.
%e A259606 RELATED SERIES.
%e A259606 A(x)^2 = x^2 + 2*x^3 + 13*x^4 + 132*x^5 + 1736*x^6 + 27276*x^7 + 490408*x^8 + 9831498*x^9 + 216085602*x^10 + ...
%e A259606 A'(x)*A(x)^2 = x^2 + 4*x^3 + 35*x^4 + 434*x^5 + 6664*x^6 + 119072*x^7 + 2392326*x^8 + 52919810*x^9 + 1271042344*x^10 + ...
%t A259606 a[n_] := Module[{A = x}, Do[A = InverseSeries[x - D[A, x] A^2 + x O[x]^n, x], {n}]; SeriesCoefficient[A, {x, 0, n}]];
%t A259606 Array[a, 25] (* _Jean-François Alcover_, Sep 28 2020, after PARI *)
%o A259606 (PARI) {a(n) = local(A=x); for(i=1,n,A=serreverse(x - A^2*A' +x*O(x^n))); polcoeff(A,n)}
%o A259606 for(n=1,25,print1(a(n),", "))
%o A259606 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A259606 {a(n)=local(A=x, B=x^2); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, (A')^m*A^(2*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
%o A259606 for(n=1, 25, print1(a(n), ", "))
%o A259606 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A259606 {a(n)=local(A=x, B=x^2); for(i=1, n, B=intformal(2*A); A = x*exp(sum(m=1, n, Dx(m-1, (A')^m*A^(2*m)/(m!*x))) +O(x^(n+1)))); polcoeff(A, n)}
%o A259606 for(n=1, 25, print1(a(n), ", "))
%Y A259606 Cf. A360578, A375456.
%K A259606 nonn
%O A259606 1,3
%A A259606 _Paul D. Hanna_, Jul 02 2015