OFFSET
1,3
COMMENTS
Absolute values are expansion of e.g.f. cosh(arcsin(x)).
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..100
FORMULA
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);
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
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
a(n) ~ (-1)^(n+1) * sinh(Pi/2) * 2^(2*n-2) * n^(2*n-3) / exp(2*n). - Vaclav Kotesovec, Oct 23 2013
For n>1, a(n) = (-1)^(n+1) * A277354(n-2). - Vaclav Kotesovec, Oct 10 2016
EXAMPLE
cos(arcsinh(x)) = 1 - x^2/2 + 5x^4/4! - 85x^6/6! + 3145x^8/8! - ...
MAPLE
seq(coeff(series(factorial(n)*cos(arcsinh(x)), x, n+1), x, n), n=0..40, 2); # Muniru A Asiru, Jul 22 2018
MATHEMATICA
Table[n!*SeriesCoefficient[Cos[ArcSinh[x]], {x, 0, n}], {n, 0, 40, 2}] (* Vaclav Kotesovec, Oct 23 2013 *)
Flatten[{1, Table[(-1)^(n+1)*Product[4*k^2 + 1, {k, 1, n}], {n, 0, 12}]}] (* Vaclav Kotesovec, Oct 10 2016 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Ralf Stephan, Dec 28 2004
STATUS
approved