Second hint to puzzle 106: Flying cards

There are 2⁵² equally likely configurations of face up/face down.  We seek the number of configurations for which the sum of face up card values is divisible by 13.

Consider the generating function
f(x) = (1 + x)⁴(1 + x²)⁴...(1 + x¹³)⁴ = a₀ + a₁x + a₂x² ... + a₃₆₄x³⁶⁴.

Each exponent in the generating function represents a total score, and the corresponding coefficient represents the number of ways of obtaining that score.

Hence we seek the sum S = a₀ + a₁₃ + ... + a₃₆₄.
Let w be a (complex) primitive 13th root of unity.  Then w¹³ = 1 and 1 + w + w² + ... + w¹² = 0.