0 1 2 1 2 3 4 55555555555555555555 4 6 7 3 48 55555 4 74 3 4 8 555555 4 1 2 1 2 1 2 1 3 21 2 19 2 1 2 1 2 0 4 0 1 2 1 2 1 2 1 0 2 1 21 2 1 21 3 3 2 3 1 2 1 0 2 3 1 2 1 2 1 3 21 2 1 2 3 19 2 00 A 4 0 0 1 A2B 0 A A 1 1 1 0 1 2 0 X X X 1 1 X X 1 X 1 X 1 2 4 00 1 x n n k 0 n k x k C 1 x n n k 0 Product form or Generating form n k x k Additive form or Expanded form 1 x n n k 0 Closed form or Generating form k 0 n k x k n k x k Since n k 0 if k n Power series Taylor series expansion 1 x n n k 0 n k x k x 1 2 n n k 0 k n The number of subsets of an n element set 0 4 1 x n n k 0 k n xk 4 0 n k 0 n 1 k k DE n k even n k n k odd n k 2n 1 0 F 0 1 x n n k 0 k n 0 F xk 4 0 n k 0 n k 1 k DE n k even n k n k odd n k 2 n 1 1 x n n k 0 n k x k 00 0 C 4 0 1 x n n k 0 0 n k x 0 H G k 0 G H n n k even k C D n n k odd k 0 0 2n 1 C D 0 0 0 0 0 C D 0 0 0 0 0 0 0 0 C D Let fn be the function that takes an n bit string and flips all its bits 0 0 0 D 0I 0B 0 0 0 0 4 0 1 x n n k 0 n k x k 00 0 1 2 1 2 1 2 1 2 1 2 B 00 0 1 2 1 2 1 2 1 2 1 2 1 2 B C0 n n n n n 0 1 2 k n 1 2 1 2 1 2 1 2 1 2 B 0 0 0 1 2 1 2 J 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 B G H 0 0 1 0 2 1 0 3 0 K L 1 1 1 1 1 2 2 1 3 1 3 3 2 1 2 1 2 3 3 3 1 M M 0 K D M D B 1 1 1 1 1 1 1 6 2 4 6 D 1 4 10 20 1 3 10 15 4 0 1 3 5 G 1 5 15 1 6 1 H Summing The Rows 2 n n 1 1 k 0 k 1 1 2 1 2 1 4 1 3 3 1 8 1 4 6 4 1 16 1 5 10 10 5 1 32 n 1 6 15 20 15 6 1 64 1 1 1 1 1 1 1 2 1 3 4 5 6 1 3 6 4 10 15 B B 1 10 20 1 5 15 1 6 1 1 1 1 1 1 1 6 1 2 3 1 4 5 10 15 1 3 6 n 10 20 i k 1 4 i i k 1 5 15 1 6 n 1 i n 1 n n 1 1 2 2 1 1 1 1 1 1 6 1 2 3 1 4 5 10 15 1 3 6 n 1 4 10 20 i k 1 5 15 1 6 1 i n 1 k k 1 1 1 1 1 2 1 N 1 3 3 1 1 1 6 1 4 5 10 15 6 4 10 20 1 5 15 1 6 1 1 1 1 1 1 1 6 1 2 3 1 4 5 10 15 1 3 6 1 4 10 20 1 5 15 1 6 1 Al Karaji Squares 1 1 1 1 1 1 1 0 1 1 2 2 1 4 3 2 3 4 2 6 5 2 10 6 2 15 20 1 9 4 1 10 5 15 16 1 6 25 1 36 0 0 00 O 0 10 5 1 1 3 1 4 1 1 5 6 n k 1 2 1 3 6 10 15 1 1 1 4 n 1 k 1 6 15 20 1 5 10 n 1 k 1 1 3 1 4 1 1 5 6 1 2 1 3 6 k even k 1 1 5 10 20 n 1 4 10 15 1 15 2 n 1 6 1 1 3 1 4 1 1 5 1 2 15 n 6 k 0 1 2 1 4 1 5 10 20 n k 1 3 10 6 1 15 2n n 6 0 1 1 2 n 1 if k 0 k n k 0 if k 0 or k n 1 1 3 1 4 1 1 5 6 n i 1 i k n 1 k 1 1 2 1 3 6 10 15 1 1 1 5 10 20 1 4 15 6 Corollary k 1 n i 1 i n 1 2 n n 1 2 n 1 K i 2 12 22 32 L2 n2 i 0 1 0 1 2 1 2 3 2 3 1 0 2 1 3 2 n n 1 n n 1 n n i i 1 2 2 3 4 5 2 2 2 2 n 2 n 1 n 1 2n 1 n 1 n 2 3 2 6 n 1 2 0 1 D D 2 C 0 0 0 P Q Q Q Q Q Q R P R 0 0 P R P R P R P R P R P R P R PM R O 1 O 1 P 2 0 R 2 P R D4 PBRJ O 1 O 1 O 1 P 2 P RJ 2 P RJ 2 P RJ PB R P B R P B R P B R B R D4 P RJ P R P R P R P R O 1 0 2J R P R P 0 R P P R P R R O 1 2 1 1 2 2 D4 J P RJ O 1 0 2J D4 J P RJ O 1 0 2J 1 2J D4 P RJ 1 2 O 1 0 2J 1 D 00 4 0 0 2 0 0 4 D 4 00 0 0 1 00 4 0 0 2 n n 1 1 k 0 2 n n 1 X k 0 00 4 …
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