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E.g.f.: Sum_{n>=0} x^n * exp( x*exp(n*x) ) / n!.
1

%I #11 Jan 29 2021 15:40:59

%S 1,2,4,14,76,532,4534,46370,559976,7845848,125535274,2265367678,

%T 45665532796,1020237666788,25091295382430,675201440266298,

%U 19774424397547216,627298393163258800,21464382547813040722,789234852942189435638,31079114556571478567396

%N E.g.f.: Sum_{n>=0} x^n * exp( 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 = x, and r = exp(x).

%e E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 76*x^4/4! + 532*x^5/5! + 4534*x^6/6! + 46370*x^7/7! + 559976*x^8/8! + 7845848*x^9/9! + ...

%e where

%e A(x) = exp(x) + x*exp(x*exp(x)) + x^2*exp(x*exp(2*x))/2! + x^3*exp(x*exp(3*x))/3! + x^4*exp(x*exp(4*x))/4! + x^5*exp(x*exp(5*x))/5! + ...

%o (PARI) {a(n) = my(A=1); A = sum(m=0, n, (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. A340332, A340335.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 29 2021