In A223511-A223522, there is a consistent missing ) for the simplify function.
For example b[j]:=simplify(x^14*diff(b[j-1], x$1);
should be b[j]:=simplify(x^14*diff(b[j-1], x$1));
That corrects the syntax, but the code doesn't do anything useful for me.
For example A223516 makes output like this (converted to 1-D):
x^14*diff(f(x),x)
x^27*(diff(diff(f(x),x),x)*x+14*diff(f(x),x))
x^40*(diff(diff(diff(f(x),x),x),x)*x^2+42*diff(diff(f(x),x),x)*x+378*diff(f(x),x))
...
Brendan.