stylistic:tone
Parity

The parity of a Stirling number of the second kind is same as the parity of a related binomial coefficient:

This relation is specified by mapping n and k coordinates onto the Sierpiński triangle.

More directly, let two sets contain positions of 1's in binary representations of results of respective expressions:

One can mimic a bitwise AND operation by intersecting these two sets:

to obtain the parity of a Stirling number of the second kind in O (1) time. In pseudocode:

where {\displaystyle \left} is the Iverson bracket.

The parity of a central Stirling number of the second kind { 2 n n } {\displaystyle \textstyle \left\{{2n \atop n}\right\}} is odd if and only if n {\displaystyle n} is a fibbinary number, a number whose binary representation has no two consecutive 1s.