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%I #10 Jan 13 2021 11:55:16
%S 1,2,7,43,441,7241,185233,7252337,429318529,38079107713,5026601726721,
%T 982190576713985,282875400939199489,119655020273311783937,
%U 74116220750762851756033,67059149595727519091988481,88432643551731676843100438529
%N E.g.f.: Sum_{n>=0} x^n * exp(2^n*x).
%C The e.g.f. A(x) of this sequence is motivated by the following identity:
%C Sum_{n>=0} p^n * exp(q*r^n) = Sum_{n>=0} (q^n/n!) / (1 - p*r^n) ;
%C here, p = x, q = x, and r = 2.
%F E.g.f.: Sum_{n>=0} x^n * exp(2^n*x).
%F E.g.f.: Sum_{n>=0} (x^n/n!) / (1 - 2^n*x).
%e E.g.f.: A(x) = 1 + 2*x + 7*x^2/2! + 43*x^3/3! + 441*x^4/4! + 7241*x^5/5! + 185233*x^6/6! + 7252337*x^7/7! + 429318529*x^8/8! + 38079107713*x^9/9! + ...
%e where
%e A(x) = exp(x) + x*exp(2*x) + x^2*exp(2^2*x) + x^3*exp(2^3*x) + x^4*exp(2^4*x) + x^5*exp(2^5*x) + ...
%e also
%e A(x) = 1/(1 - x) + x/(1 - 2*x) + (x^2/2!)/(1 - 2^2*x) + (x^3/3!)/(1 - 2^3*x) + (x^4/4!)/(1 - 2^4*x) + (x^5/5!)/(1 - 2^5*x) + ...
%o (PARI) {a(n) = my(A = sum(m=0,n, x^m * exp(2^m*x +x*O(x^n)) )); n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n) = my(A = sum(m=0,n, x^m/m! * 1/(1 - 2^m*x +x*O(x^n)) )); n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A340450.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 11 2021
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