Perl 6, 114 bytes

my @b=1;for 1..+slurp() {@b.push: 1-[+] (^$^m).map: {([*] $m+1-$_..$m)*@b[$_]/($m+1-$_)/([*] 1..$_)}};say @b[*-1];

I could have made it shorter, but that would sacrifice lots of speed.

Perl 6, 83 bytes

my &B={$^m??1-[+] (^$m).map: {combinations($m,$_)*B($_)/($m+1-$_)}!!1};say B slurp;

A faster, 114-byte solution:

my @b=1;for 1..+slurp() {@b.push: 1-[+] (^$^m).map: {([*] $m+1-$_..$m)*@b[$_]/($m+1-$_)/([*] 1..$_)}};say @b[*-1];