|
%I #11 Jul 22 2018 08:46:38
%S 1,4,128,43680,178946048,9382409745280,6558834518571089920,
%T 62879485860387254833099776,8439542720341303996200869198561280,
%U 16110026846830031883594370688522189192189952
%N G.f.: Sum_{n>=0} arcsinh(4^n*x)^n/n!, a power series in x having only integer coefficients.
%H G. C. Greubel, <a href="/A141367/b141367.txt">Table of n, a(n) for n = 0..40</a>
%F a(n) = [x^n] [ sqrt(1+x^2) + x ]^(4^n).
%F More generally, the following coefficient of x^n in the series:
%F [x^n] Sum_{n>=0} arcsinh(q^n*x)^n/n! = [x^n] [ sqrt(1+x^2) + x ]^(q^n) is an integer for any even integer q.
%e G.f.: A(x) = 1 + 4*x + 128*x^2 + 43680*x^3 + 178946048*x^4 + ...
%t Table[SeriesCoefficient[(Sqrt[1 + x^2] + x)^(4^n), {x, 0, n}], {n, 0, 25}] (* _G. C. Greubel_, Apr 15 2017 *)
%o (PARI) {a(n)=polcoeff(sum(k=0,n, asinh(4^k*x +x*O(x^n))^k/k!),n)}
%o (PARI) {a(n)=polcoeff((x+sqrt(1+x^2 +x*O(x^n)))^(4^n),n)}
%Y Cf. A136647, A141368.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 02 2008
| 