A368175 Square array read by ascending antidiagonals: T(n,k) = Sum_{i=ceiling((k-n)/2)..floor((k+n-1)/2)} binomial(k,i), with n >= 1, k >= 0.
1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 1, 2, 4, 6, 6, 1, 2, 4, 7, 10, 10, 1, 2, 4, 8, 14, 20, 20, 1, 2, 4, 8, 15, 25, 35, 35, 1, 2, 4, 8, 16, 30, 50, 70, 70, 1, 2, 4, 8, 16, 31, 56, 91, 126, 126, 1, 2, 4, 8, 16, 32, 62, 112, 182, 252, 252, 1, 2, 4, 8, 16, 32, 63, 119, 210, 336, 462, 462
Offset: 1
Examples
Array begins: n\k| 0 1 2 3 4 5 6 7 8 9 10 ... ---+-------------------------------------------------- 1 | 1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, ... = A001405 2 | 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, ... = A001405 3 | 1, 2, 4, 7, 14, 25, 50, 91, 182, 336, 672, ... = A026010 4 | 1, 2, 4, 8, 15, 30, 56, 112, 210, 420, 792, ... = A026023 5 | 1, 2, 4, 8, 16, 31, 62, 119, 238, 456, 912, ... 6 | 1, 2, 4, 8, 16, 32, 63, 126, 246, 492, 957, ... 7 | 1, 2, 4, 8, 16, 32, 64, 127, 254, 501, 1002, ... 8 | 1, 2, 4, 8, 16, 32, 64, 128, 255, 510, 1012, ... 9 | 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1022, ... 10 | 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, ... ... For n = 3 and k = 4 the 14 members of S are 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110.
References
- Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, exercises 71 and 72, pp. 479 and 799.
Links
- Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened).
Comments