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%I #12 Aug 19 2018 03:26:32
%S 1,4,128,43712,178978816,9382678180864,6558857974821945344,
%T 62879510456046477909016576,8439543050458648574249946721550336,
%U 16110026905906831711301708576024644666261504
%N G.f.: Sum_{n>=0} arctanh(4^n*x)^n/n!, a power series in x having only integer coefficients.
%H G. C. Greubel, <a href="/A141368/b141368.txt">Table of n, a(n) for n = 0..40</a>
%F a(n) = [x^n] [ sqrt((1+x)/(1-x)) ]^(4^n).
%F More generally, the following coefficient of x^n in the series:
%F [x^n] Sum_{n>=0} arctanh(q^n*x)^n/n! = [x^n] [ sqrt((1+x)/(1-x)) ]^(q^n) is an integer for any even integer q.
%e G.f. A(x) = 1 + 4*x + 128*x^2 + 43712*x^3 + 178978816*x^4 + ...
%t Table[SeriesCoefficient[(Sqrt[(1 + x)/(1 - x)])^(4^n), {x, 0, n}], {n, 0, 25}] (* _G. C. Greubel_, Apr 15 2017 *)
%o (PARI) {a(n)=polcoeff(sum(k=0,n,atanh(4^k*x +x*O(x^n))^k/k!),n)}
%o (PARI) {a(n)=polcoeff(((1+x)/(1-x +x*O(x^n)))^(4^n/2),n)}
%Y Cf. A136749, A141367.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 02 2008
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