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A360578 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion( x - x*A'(x)*A(x) ).

%I A360578 #20 Feb 22 2023 12:26:38
%S A360578 1,1,5,42,471,6422,101439,1803949,35459549,760744865,17651187689,
%T A360578 439893743313,11711735210140,331666197753372,9954249177284539,
%U A360578 315638779480717743,10545365878475964736,370309787453143694246,13637805276205022293179,525684316153586923528166
%N A360578 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion( x - x*A'(x)*A(x) ).
%H A360578 Paul D. Hanna, <a href="/A360578/b360578.txt">Table of n, a(n) for n = 1..300</a>
%F A360578 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
%F A360578 (1) A( x - x*A'(x)*A(x) ) = x.
%F A360578 (2) A(x) = x + A(x) * A'(A(x)) * A(A(x)).
%F A360578 (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * A'(x)^n * A(x)^n / n!.
%F A360578 (4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * A'(x)^n * A(x)^n / n! ).
%F A360578 a(n) ~ c * n! * n^alfa / LambertW(1)^n, where alfa = 1.5447806483693... and c = 0.02888888614196289496..., conjecture: alfa = 2*(2*LambertW(1) - 1 + 1/(1 + LambertW(1))). - _Vaclav Kotesovec_, Feb 22 2023
%e A360578 G.f.: A(x) = x + x^2 + 5*x^3 + 42*x^4 + 471*x^5 + 6422*x^6 + 101439*x^7 + 1803949*x^8 + 35459549*x^9 + 760744865*x^10 + ...
%e A360578 such that A( x - x*A'(x)*A(x) ) = x.
%e A360578 Related series.
%e A360578 Series_Reversion(A(x)) = x - x^2 - 3*x^3 - 22*x^4 - 235*x^5 - 3153*x^6 - 49721*x^7 - 888784*x^8 - 17615520*x^9 + ...
%e A360578 A'(x)*A(x) = x + 3*x^2 + 22*x^3 + 235*x^4 + 3153*x^5 + 49721*x^6 + 888784*x^7 + 17615520*x^8 + ...
%e A360578 A(A(x)) = x + 2*x^2 + 12*x^3 + 110*x^4 + 1294*x^5 + 18127*x^6 + 290620*x^7 + 5206800*x^8 + 102633591*x^9 + ...
%e A360578 A'(A(x)) = 1 + 2*x + 17*x^2 + 208*x^3 + 3108*x^4 + 53328*x^5 + 1018948*x^6 + 21297818*x^7 + 481458997*x^8 + ...
%e A360578 A'(A(x))*A(A(x)) = x + 4*x^2 + 33*x^3 + 376*x^4 + 5242*x^5 + 84625*x^6 + 1534652*x^7 + 30682881*x^8 + ...
%o A360578 (PARI) {a(n) = my(A=x); for(i=1, n, A=serreverse(x - x*A'*A +x*O(x^n))); polcoeff(A, n)}
%o A360578 for(n=1, 25, print1(a(n), ", "))
%o A360578 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A360578 {a(n) = my(A=x); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, x^m*(A')^m*A^m/m!)) +O(x^(n+1))); polcoeff(A, n)}
%o A360578 for(n=1, 25, print1(a(n), ", "))
%o A360578 (PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
%o A360578 {a(n)=local(A=x); for(i=1, n, A = x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*(A')^m*A^m/m!)) +O(x^(n+1)))); polcoeff(A, n)}
%o A360578 for(n=1, 25, print1(a(n), ", "))
%Y A360578 Cf. A259606, A213591, A303063.
%K A360578 nonn
%O A360578 1,3
%A A360578 _Paul D. Hanna_, Feb 21 2023