Exact comparison of three methods
A(n,m) = sum((-1)^k * C(n,k)^2 * k^m)
Each method: 5 timed runs; median is reported.
The direct sum updates C(n,k) incrementally.

n=100, m=10
Our recurrence:      0.000013 s
Krawtchouk method:   0.000057 s
Direct sum:          0.000030 s
Exact results match: yes
Direct / our recurrence: 2.31 times
Krawtchouk / our recurrence: 4.47 times

n=1000, m=20
Our recurrence:      0.000061 s
Krawtchouk method:   0.006155 s
Direct sum:          0.001214 s
Exact results match: yes
Direct / our recurrence: 19.95 times
Krawtchouk / our recurrence: 101.16 times

n=10000, m=20
Our recurrence:      0.001898 s
Krawtchouk method:   0.376118 s
Direct sum:          0.216951 s
Exact results match: yes
Direct / our recurrence: 114.33 times
Krawtchouk / our recurrence: 198.21 times

n=10000, m=50
Our recurrence:      0.002197 s
Krawtchouk method:   2.335512 s
Direct sum:          0.247151 s
Exact results match: yes
Direct / our recurrence: 112.48 times
Krawtchouk / our recurrence: 1062.91 times

Exhaustive check passed for 0 <= n <= 30 and 0 <= m <= 15.