A306680 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^(k+1)).

1, 1, 2, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 3, 5, 1, 1, 1, 2, 5, 6, 1, 1, 1, 1, 4, 8, 7, 1, 1, 1, 1, 2, 7, 13, 8, 1, 1, 1, 1, 1, 5, 12, 21, 9, 1, 1, 1, 1, 1, 2, 11, 21, 34, 10, 1, 1, 1, 1, 1, 1, 6, 21, 37, 55, 11, 1, 1, 1, 1, 1, 1, 2, 16, 37, 65, 89, 12

Offset

0,3

Links

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Formula

A(n,k) = Sum_{j=0..n} binomial(n-j,k*j).

A(n,k) = A306713(k*n,k) for k > 0.

Example

A(4,1) = A306713(4,1) = 5, A(4,2) = A306713(8,2) = 4.
Square array begins:
   1,  1,  1,  1,  1,  1, 1, 1, 1, ...
   2,  1,  1,  1,  1,  1, 1, 1, 1, ...
   3,  2,  1,  1,  1,  1, 1, 1, 1, ...
   4,  3,  2,  1,  1,  1, 1, 1, 1, ...
   5,  5,  4,  2,  1,  1, 1, 1, 1, ...
   6,  8,  7,  5,  2,  1, 1, 1, 1, ...
   7, 13, 12, 11,  6,  2, 1, 1, 1, ...
   8, 21, 21, 21, 16,  7, 2, 1, 1, ...
   9, 34, 37, 37, 36, 22, 8, 2, 1, ...

Mathematica

T[n_, k_] := Sum[Binomial[n - j, k*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

Crossrefs

Columns 0-9 give A000027(n+1), A000045(n+1), A005251(n+1), A003522, A005676, A099132, A293169, A306721, A306752, A306753.

Cf. A306646, A306713, A306735.

Sequence in context: A205131 A175892 A010783 * A083312 A032435 A337798

Adjacent sequences: A306677 A306678 A306679 * A306681 A306682 A306683

Keyword

nonn,tabl,look

Author

Seiichi Manyama, Mar 05 2019

Status

approved