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E.g.f. cos(arcsinh(x)) = sin(arccosh(x)) (even powers only).
5

%I #30 Jul 22 2018 08:45:07

%S 1,-1,5,-85,3145,-204425,20646925,-2993804125,589779412625,

%T -151573309044625,49261325439503125,-19753791501240753125,

%U 9580588878101765265625,-5527999782664718558265625,3742455852864014463945828125,-2937827844498251354197475078125,2646982887892924470131925045390625

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

%C Absolute values are expansion of e.g.f. cosh(arcsin(x)).

%H Muniru A Asiru, <a href="/A101928/b101928.txt">Table of n, a(n) for n = 1..100</a>

%F 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 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 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 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 For n>1, a(n) = (-1)^(n+1) * A277354(n-2). - _Vaclav Kotesovec_, Oct 10 2016

%e cos(arcsinh(x)) = 1 - x^2/2 + 5x^4/4! - 85x^6/6! + 3145x^8/8! - ...

%p seq(coeff(series(factorial(n)*cos(arcsinh(x)), x,n+1),x,n),n=0..40,2); # _Muniru A Asiru_, Jul 22 2018

%t Table[n!*SeriesCoefficient[Cos[ArcSinh[x]],{x,0,n}],{n,0,40,2}] (* _Vaclav Kotesovec_, Oct 23 2013 *)

%t Flatten[{1, Table[(-1)^(n+1)*Product[4*k^2 + 1, {k, 1, n}], {n, 0, 12}]}] (* _Vaclav Kotesovec_, Oct 10 2016 *)

%Y Bisection of A006228.

%Y Cf. A079484, A277354.

%K sign

%O 1,3

%A _Ralf Stephan_, Dec 28 2004