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A101928 E.g.f. cos(arcsinh(x)) = sin(arccosh(x)) (even powers only).

%I A101928 #30 Jul 22 2018 08:45:07
%S A101928 1,-1,5,-85,3145,-204425,20646925,-2993804125,589779412625,
%T A101928 -151573309044625,49261325439503125,-19753791501240753125,
%U A101928 9580588878101765265625,-5527999782664718558265625,3742455852864014463945828125,-2937827844498251354197475078125,2646982887892924470131925045390625
%N A101928 E.g.f. cos(arcsinh(x)) = sin(arccosh(x)) (even powers only).
%C A101928 Absolute values are expansion of e.g.f. cosh(arcsin(x)).
%H A101928 Muniru A Asiru, <a href="/A101928/b101928.txt">Table of n, a(n) for n = 1..100</a>
%F A101928 E.g.f.: cos(arcsinh(x)) =  sqrt(1+x^2)*(1-x^2*(1-5*x^2/(G(0)+5*x^2))); G(k) = (k+2)*(2*k+3)-x^2*(2*k^2+6*k+5)+x^2*(k+2)*(2*k+3)*(2*k^2+10*k+13)/G(k+1);
%F A101928 For cosh(arcsin(x)) = sqrt(1-x^2)*(1 + x^2*(1 + 5*x^2/(G(0) - 5*x^2))); G(k) = x^2*(2*k^2+6*k+5) + (k+2)*(2*k+3) - x^2*(k+2)*(2*k+3)*(2*k^2+10*k+13)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 19 2011
%F A101928 G.f.: 1 - x*(1 + x*(G(0) - 1)/(x-1)) where G(k) = 1 + ((2*k+2)^2+1)/(1-x/(x - 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 15 2013
%F A101928 a(n) ~ (-1)^(n+1) * sinh(Pi/2) * 2^(2*n-2) * n^(2*n-3) / exp(2*n). - _Vaclav Kotesovec_, Oct 23 2013
%F A101928 For n>1, a(n) = (-1)^(n+1) * A277354(n-2). - _Vaclav Kotesovec_, Oct 10 2016
%e A101928 cos(arcsinh(x)) = 1 - x^2/2 + 5x^4/4! - 85x^6/6! + 3145x^8/8! - ...
%p A101928 seq(coeff(series(factorial(n)*cos(arcsinh(x)), x,n+1),x,n),n=0..40,2); # _Muniru A Asiru_, Jul 22 2018
%t A101928 Table[n!*SeriesCoefficient[Cos[ArcSinh[x]],{x,0,n}],{n,0,40,2}] (* _Vaclav Kotesovec_, Oct 23 2013 *)
%t A101928 Flatten[{1, Table[(-1)^(n+1)*Product[4*k^2 + 1, {k, 1, n}], {n, 0, 12}]}] (* _Vaclav Kotesovec_, Oct 10 2016 *)
%Y A101928 Bisection of A006228.
%Y A101928 Cf. A079484, A277354.
%K A101928 sign
%O A101928 1,3
%A A101928 _Ralf Stephan_, Dec 28 2004