A306846 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-1))/((1-x)^k-x^k).

1, 1, 2, 1, 1, 4, 1, 1, 2, 8, 1, 1, 1, 4, 16, 1, 1, 1, 2, 8, 32, 1, 1, 1, 1, 5, 16, 64, 1, 1, 1, 1, 2, 11, 32, 128, 1, 1, 1, 1, 1, 6, 22, 64, 256, 1, 1, 1, 1, 1, 2, 16, 43, 128, 512, 1, 1, 1, 1, 1, 1, 7, 36, 85, 256, 1024, 1, 1, 1, 1, 1, 1, 2, 22, 72, 170, 512, 2048

Offset

0,3

Links

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

Formula

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

Example

Square array begins:
     1,   1,  1,  1,  1,  1, 1, 1, 1, ...
     2,   1,  1,  1,  1,  1, 1, 1, 1, ...
     4,   2,  1,  1,  1,  1, 1, 1, 1, ...
     8,   4,  2,  1,  1,  1, 1, 1, 1, ...
    16,   8,  5,  2,  1,  1, 1, 1, 1, ...
    32,  16, 11,  6,  2,  1, 1, 1, 1, ...
    64,  32, 22, 16,  7,  2, 1, 1, 1, ...
   128,  64, 43, 36, 22,  8, 2, 1, 1, ...
   256, 128, 85, 72, 57, 29, 9, 2, 1, ...

Mathematica

T[n_, k_] := Sum[Binomial[n, k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

Crossrefs

Columns 1-9 give A000079, A011782, A024493, A038503, A139398, A306847, A306852, A306859, A306860.

Cf. A306680.

Sequence in context: A333810 A205592 A099238 * A327315 A141450 A209690

Adjacent sequences: A306843 A306844 A306845 * A306847 A306848 A306849

Keyword

nonn,tabl

Author

Seiichi Manyama, Mar 13 2019

Status

approved