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A369551 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*exp(x) * A(x^2*exp(x)).

%I A369551 #8 Jan 29 2024 10:59:26
%S A369551 1,1,2,9,52,365,3126,33607,434120,6397785,104813290,1881831611,
%T A369551 36703128012,773468319637,17544261523166,427299522260535,
%U A369551 11158470652237456,311944165793916977,9313905287778153426,296051128664550195763,9979462242106491507860,355292353569342771519021
%N A369551 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*exp(x) * A(x^2*exp(x)).
%C A369551 Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).
%F A369551 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A369551 (1) A(x) = 1 + x*exp(x) * A(x^2*exp(x)).
%F A369551 (2) A(x) = (1/x) * Sum_{n>=0} F(n), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
%F A369551 (3) A(x) = log(G(x)) / x where G(x) = exp(x) * G(x^2*exp(x)) is the e.g.f. of A369550.
%F A369551 (4) A(x) = L(x)/x where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.
%F A369551 a(n) = A369091(n+1)/(n+1) for n >= 0.
%e A369551 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 365*x^5/5! + 3126*x^6/6! + 33607*x^7/7! + 434120*x^8/8! + 6397785*x^9/9! + 104813290*x^10/10! + ...
%e A369551 RELATED SERIES.
%e A369551 The expansion of A(x^2*exp(x)) begins
%e A369551 A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2550*x^6/6! + 29442*x^7/7! + 386456*x^8/8! + ...
%e A369551 where A(x) = 1 + x*exp(x) * A(x^2*exp(x)).
%e A369551 The expansion of exp(x*A(x)) is the e.g.f. of A369550, which begins
%e A369551 exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! + 6901*x^6/6! + 79045*x^7/7! + 1049385*x^8/8! + ... + A369550(n)*x^n/n! + ...
%o A369551 (PARI) {a(n) = my(A=1+x, X = x + x*O(x^n)); for(i=1, n, A = 1 + x*exp(X) * subst(A, x, x^2*exp(X)) ); n!*polcoeff(A, n)}
%o A369551 for(n=0, 30, print1(a(n), ", "))
%Y A369551 Cf. A369090, A369091, A369550.
%K A369551 nonn
%O A369551 0,3
%A A369551 _Paul D. Hanna_, Jan 29 2024