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%I #5 Jan 15 2018 15:34:34
%S 1,2,16,496,63488,32899840,68049141760,560546415810560,
%T 18415229458563727360,2416302337337071616327680,
%U 1267360474688679165942982246400,2658246833688954938616062542151680000
%N E.g.f.: Sum_{n>=0} arctan(2^n*x)^n/n!.
%F a(n) = [x^n/n!] exp(2^n*arctan(x)) for n >= 0.
%e E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 496*x^3/3! + 63488*x^4/4! + ...
%e A(x) = 1 + arctan(2*x) + arctan(4*x)^2/2! + arctan(8*x)^3/3! + arctan(16*x)^4/4! + ... + arctan(2^n*x)^n/n! + ...
%e a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(arctan(x)):
%e G(x) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! + 5*x^5/5! + 145*x^6/6! + ... + A002019(n)*x^n/n! + ...
%o (PARI) {a(n)=n!*polcoeff(sum(k=0,n,atan(2^k*x +x*O(x^n))^k/k!),n)}
%o (PARI) {a(n)=n!*polcoeff(exp(2^n*atan(x +x*O(x^n))),n)}
%Y Cf. A002019 (exp(arctan x)), variants: A136632, A168402, A168403, A168404, A168405, A168407, A168408.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 25 2009
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