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A136647 G.f.: A(x) = Sum_{n>=0} arcsinh( 2^n*x )^n / n!; a power series in x with integer coefficients.

%I A136647 #13 Mar 16 2021 01:26:13
%S A136647 1,2,8,84,2688,276892,94978048,111457917800,457117679616000,
%T A136647 6660816097416169260,349290546231751288553472,
%U A136647 66597307693046550483175282456,46556113319179632622352835689840640
%N A136647 G.f.: A(x) = Sum_{n>=0} arcsinh( 2^n*x )^n / n!; a power series in x with integer coefficients.
%H A136647 G. C. Greubel, <a href="/A136647/b136647.txt">Table of n, a(n) for n = 0..50</a>
%F A136647 a(n) = [y^n] ( sqrt(1+y^2) + y )^(2^n), since log(sqrt(1+y^2) + y) = arcsinh(y); [y^n] F(y) denotes the coefficient of y^n in F(y).
%e A136647 G.f.: A(x) = 1 + 2*x + 8*x^2 + 84*x^3 + 2688*x^4 + 276892*x^5 + ...
%e A136647 This is a special application of the following identity.
%e A136647 Let F(x), G(x), be power series in x such that F(0)=1, G(0)=1, then
%e A136647 Sum_{n>=0} m^n * H(q^n*x) * log( F(q^n*x)*G(x) )^n / n! =
%e A136647 Sum_{n>=0} x^n * G(x)^(m*q^n) * [y^n] H(y)*F(y)^(m*q^n).
%p A136647 m:=30;
%p A136647 S:=series( add( arcsinh(2^j*x)^j/j! , j=0..m+2), x, m+1):
%p A136647 seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Mar 15 2021
%t A136647 With[{m=30}, CoefficientList[Series[Sum[ArcSinh[2^j*x]^j/j!, {j,0,m+2}], {x,0,m}], x]] (* _G. C. Greubel_, Mar 15 2021 *)
%o A136647 (PARI) {a(n)=polcoeff(sum(k=0,n, asinh(2^k*x +x*O(x^n))^k/k!),n)}
%o A136647 (PARI) {a(n)=polcoeff((sqrt(1+x^2)+x+x*O(x^n))^(2^n),n)}
%o A136647 (Magma)
%o A136647 m:=30;
%o A136647 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A136647 Coefficients(R!( (&+[Argsinh(2^j*x)^j/Factorial(j): j in [0..m+2]]) )); // _G. C. Greubel_, Mar 15 2021
%Y A136647 Cf. A136558.
%K A136647 nonn
%O A136647 0,2
%A A136647 _Paul D. Hanna_, Jan 20 2008