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A369550 Expansion of e.g.f. A(x) satisfying A(x) = exp(x) * A(x^2*exp(x)).

%I A369550 #13 Jan 29 2024 11:02:08
%S A369550 1,1,3,13,85,701,6901,79045,1049385,15924025,271248121,5108389001,
%T A369550 105158055949,2346022349269,56348945801877,1449434215375021,
%U A369550 39758549273200081,1159092552400164977,35813081725133941297,1169791166246561367697,40297553373717279300981,1460613225168596836153741
%N A369550 Expansion of e.g.f. A(x) satisfying A(x) = exp(x) * A(x^2*exp(x)).
%C A369550 Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).
%F A369550 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A369550 (1) A(x) = exp(x) * A(x^2*exp(x)).
%F A369550 (2) A(x) = exp( Sum_{n>=0} F(n) ), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
%F A369550 (3) A(x) = exp(L(x)) where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.
%F A369550 (4) A(x) = G(x)/x where G(x) = G(x^2*exp(x))/x is the e.g.f. of A369090.
%F A369550 a(n) = A369090(n+1)/(n+1) for n >= 0.
%e A369550 E.g.f.: 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! + 15924025*x^9/9! + ...
%e A369550 RELATED SERIES.
%e A369550 The expansion of A(x^2*exp(x)) begins
%e A369550 exp(-x) * A(x) = A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 380*x^5/5! + 3750*x^6/6! +  + 42882*x^7/7! + 576296*x^8/8! + ...
%e A369550 The logarithm of e.g.f. A(x) equals L(x) where L(x) = x + L(x^2*exp(x)),
%e A369550 L(x) = x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2190*x^6/6! + 21882*x^7/7! + 268856*x^8/8! + ... + A369091(n)*x^n/n! + ...
%o A369550 (PARI) {a(n) = my(A=1+x, X = x + x*O(x^n)); for(i=1,n, A = exp(X) * subst(A,x,x^2*exp(X)) ); n!*polcoeff(A,n)}
%o A369550 for(n=0,30, print1(a(n),", "))
%Y A369550 Cf. A369090, A369091, A369551.
%K A369550 nonn
%O A369550 0,3
%A A369550 _Paul D. Hanna_, Jan 29 2024