%I #5 Feb 01 2021 13:22:02
%S 1,3,9,39,249,2083,21309,258891,3676465,60188355,1120986549,
%T 23477257147,547630675689,14116139238243,399459984039853,
%U 12338912081665707,413926973626402401,15012208323417943171,586241726429900170341,24560642734838666902107
%N E.g.f.: Sum_{n>=0} x^n * exp( 2*x*exp(n*x) ) / n!.
%C The e.g.f. A(x) of this sequence is motivated by the following identity:
%C Sum_{n>=0} p^n/n! * exp(q*r^n) = Sum_{n>=0} q^n/n! * exp(p*r^n) ;
%C here, p = x, q = 2*x, and r = exp(x).
%F E.g.f.: Sum_{n>=0} x^n * exp( 2*x*exp(n*x) ) / n!.
%F E.g.f.: Sum_{n>=0} 2^n*x^n * exp( x*exp(n*x) ) / n!.
%e E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 39*x^3/3! + 249*x^4/4! + 2083*x^5/5! + 21309*x^6/6! + 258891*x^7/7! + 3676465*x^8/8! + 60188355*x^9/9! + ...
%e where
%e A(x) = exp(2*x) + x*exp(2*x*exp(x)) + x^2*exp(2*x*exp(2*x))/2! + x^3*exp(2*x*exp(3*x))/3! + x^4*exp(2*x*exp(4*x))/4! + x^5*exp(2*x*exp(5*x))/5! + ...
%e also
%e A(x) = exp(x) + 2*x*exp(x*exp(x)) + 2^2*x^2*exp(x*exp(2*x))/2! + 2^3*x^3*exp(x*exp(3*x))/3! + 2^4*x^4*exp(x*exp(4*x))/4! + 2^5*x^5*exp(x*exp(5*x))/5! + ...
%o (PARI) {a(n) = my(A=1); A = sum(m=0, n, (x^m/m!)*exp(2*x*exp(m*x +x*O(x^n)))); n!*polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=1); A = sum(m=0, n, (2^m*x^m/m!)*exp(x*exp(m*x +x*O(x^n)))); n!*polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A340909, A340332, A340335.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 30 2021