OFFSET
0,3
FORMULA
a(n) = ((2*n)!*sum(k=0..n, C(n-1,n-k)/(2*k)!*(-1)^(n-k))). - Vladimir Kruchinin, Jun 17 2011
E.g.f.: cosh(x/sqrt(1+x^2)) = 1 + x^2/(G(0) - x^2) where G(k)= (2*k+2)*(2*k+1)*(1+x^2) + x^2 - (2*k+2)*(2*k+1)*x^2*(1+x^2)/G(k+1); (continued fraction, Euler's kind, 1-step). - Sergei N. Gladkovskii, Aug 06 2012
D-finite with recurrence: a(n) = -(12*n^2 - 24*n + 11)*a(n-1) - 12*(n-2)*(n-1)*(2*n-3)^2*a(n-2) - 16*(n-3)*(n-2)^2*(n-1)*(2*n-5)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Nov 09 2013
Lim sup n->infinity |a(n)|/(2^(2*n+2/3) * exp(3/4*(2*n)^(1/3)-2*n) * n^(2*n-1/3) / sqrt(3)) = 1. - Vaclav Kotesovec, Nov 09 2013
EXAMPLE
cosh(sin(arctan(x))) = 1+1/2!*x^2-11/4!*x^4+301/6!*x^6-15287/8!*x^8...
MATHEMATICA
Table[n!*SeriesCoefficient[Cosh[x/Sqrt[1+x^2]], {x, 0, n}], {n, 0, 40, 2}] (* Vaclav Kotesovec, Nov 08 2013 *)
PROG
(Maxima)
a(n):=((2*n)!*sum(binomial(n-1, n-k)/(2*k)!*(-1)^(n-k), k, 0, n)); [Vladimir Kruchinin, Jun 17 2011]
CROSSREFS
KEYWORD
sign
AUTHOR
Patrick Demichel (patrick.demichel(AT)hp.com)
STATUS
approved