Hickerson series of almost integers

func h(n) {
    n! / (2 * pow(2.log, n+1))
}
 
{ |n|
    "h(%2d) = %22s is%s almost an integer.\n".printf(
        n, var hn = h(n).round(-3), hn.to_s ~~ /\.[09]/ ? '' : ' NOT')
} << 1..17

Output:

h( 1) =                  1.041 is almost an integer.
h( 2) =                  3.003 is almost an integer.
h( 3) =                 12.996 is almost an integer.
h( 4) =                 74.999 is almost an integer.
h( 5) =                541.002 is almost an integer.
h( 6) =               4683.001 is almost an integer.
h( 7) =              47292.999 is almost an integer.
h( 8) =             545834.998 is almost an integer.
h( 9) =            7087261.002 is almost an integer.
h(10) =          102247563.005 is almost an integer.
h(11) =         1622632572.998 is almost an integer.
h(12) =        28091567594.982 is almost an integer.
h(13) =       526858348381.001 is almost an integer.
h(14) =     10641342970443.085 is almost an integer.
h(15) =    230283190977853.037 is almost an integer.
h(16) =   5315654681981354.513 is NOT almost an integer.
h(17) = 130370767029135900.458 is NOT almost an integer.