Counting valid Binary Sudoku rows
Jelly, &nbsp;13 12&nbsp; 9 bytes

ŻcṚ$+Ḋ$²S


Try it online! Or see the test-suite.

How?

This code is calculating

$$\sum_{k=\lceil\frac{n}{2}\rceil}^{n}\big(\binom{k}{n-k}+\binom{k+1}{n-(k+1)}\big)^2$$

(where \$k\$ starts at \$0\$ rather than \$\lceil\frac{n}{2}\rceil\$ ...the additional terms are \$0\$ but allows a reversal trick)

ŻcṚ$+Ḋ$²S - Link: integer, n       e.g. 7
Ż         - zero range                  [0,  1,  2,  3,  4,  5,  6,  7]
   $      - last two links as a monad:
  Ṛ       -   reverse                   [7,  6,  5,  4,  3,  2,  1,  0]
 c        -   n-choose-k                [0,  0,  0,  0,  4, 10,  6,  1]
      $   - last two links as a monad:
     Ḋ    -   dequeue                   [0,  0,  0,  4, 10,  6,  1]
    +     -   add                       [0,  0,  0,  4, 14, 16,  7,  1]
       ²  - square                      [0,  0,  0, 16,196,256, 49,  1]
        S - sum                         518