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

1,5

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 *)

Crossrefs

Cf. A000079, A000120, A000225, A001405, A014495, A026010, A026023, A191781, A367508.

Sequence in context: A331254 A193738 A230494 * A106580 A355080 A165915

Adjacent sequences: A368172 A368173 A368174 * A368176 A368177 A368178

Keyword

nonn,tabl,easy

Author

Paolo Xausa, Dec 14 2023

Status

approved