A366986 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} binomial(d+k-1,k).

1, 1, 2, 1, 3, 2, 1, 4, 4, 3, 1, 5, 7, 7, 2, 1, 6, 11, 14, 6, 4, 1, 7, 16, 25, 16, 12, 2, 1, 8, 22, 41, 36, 31, 8, 4, 1, 9, 29, 63, 71, 71, 29, 15, 3, 1, 10, 37, 92, 127, 147, 85, 50, 13, 4, 1, 11, 46, 129, 211, 280, 211, 145, 52, 18, 2, 1, 12, 56, 175, 331, 498, 463, 371, 176, 74, 12, 6

Offset

1,3

Links

Table of n, a(n) for n=1..78.

Formula

G.f. of column k: Sum_{j>=1} x^j/(1 - x^j)^(k+1).

Example

Square  array begins:
  1,  1,  1,  1,   1,   1,   1, ...
  2,  3,  4,  5,   6,   7,   8, ...
  2,  4,  7, 11,  16,  22,  29, ...
  3,  7, 14, 25,  41,  63,  92, ...
  2,  6, 16, 36,  71, 127, 211, ...
  4, 12, 31, 71, 147, 280, 498, ...
  2,  8, 29, 85, 211, 463, 925, ...

Prog

(PARI) T(n, k) = sumdiv(n, d, binomial(d+k-1, k));

Crossrefs

Columns k=0..5 give A000005, A000203, A007437, A059358, A073570, A101289.

T(n,n-1) gives A332508.

T(n,n) gives A343548.

Cf. A366977.

Sequence in context: A062001 A361043 A181847 * A209562 A259344 A239030

Adjacent sequences: A366983 A366984 A366985 * A366987 A366988 A366989

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 31 2023

Status

approved