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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.
3
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
1,5
COMMENTS
T(n,k), for k >= 1, is the size of the largest possible set S of k-bit strings such that, if S_a < S_b are members of S, then W(S_b) < W(S_a) + n, where W is A000120.
T(1,k), for k >= 1, gives the number of rows in the Christmas tree pattern (cf. A367508) of order k. Furthermore, T(n,k), for k >= 1, gives the number of rows generated by iteratively applying k times the map described in A367508, starting from a single row of length n.
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).
FORMULA
T(n,0) = 1.
T(1,k) = A001405(k).
T(n,k) = 2^k = A000079(k), for n > k.
T(n,n) = 2^n - 1 = A000225(n).
Antidiagonal sums: Sum_{n=1..d} T(n,d-n) = binomial(d+1,floor((d+1)/2)) - 1 = A014495(d+1), for d >= 1.
EXAMPLE
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.
MATHEMATICA
A368175[n_, k_]:=If[n>k, 2^k, Sum[Binomial[k, i], {i, Ceiling[(k-n)/2], Floor[(k+n-1)/2]}]];
With[{dmax=15}, Table[A368175[n-k, k], {n, dmax}, {k, 0, n-1}]] (* Generates 15 antidiagonals *)
KEYWORD
nonn,tabl,easy
AUTHOR
Paolo Xausa, Dec 14 2023
STATUS
approved