A345032 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} k^(floor(n/j) - 1).

1, 1, 1, 1, 2, 2, 1, 3, 3, 2, 1, 4, 6, 4, 3, 1, 5, 11, 12, 5, 3, 1, 6, 18, 32, 21, 6, 4, 1, 7, 27, 70, 87, 41, 7, 4, 1, 8, 38, 132, 263, 258, 74, 8, 5, 1, 9, 51, 224, 633, 1047, 745, 144, 9, 5, 1, 10, 66, 352, 1305, 3158, 4120, 2224, 275, 10, 6, 1, 11, 83, 522, 2411, 7821, 15659, 16460, 6605, 541, 11, 6

Offset

1,5

Links

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

Formula

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

Example

Square array begins:
  1, 1,  1,   1,    1,    1,    1, ...
  1, 2,  3,   4,    5,    6,    7, ...
  2, 3,  6,  11,   18,   27,   38, ...
  2, 4, 12,  32,   70,  132,  224, ...
  3, 5, 21,  87,  263,  633, 1305, ...
  3, 6, 41, 258, 1047, 3158, 7821, ...

Mathematica

T[n_, 0] := Floor[(n + 1)/2]; T[n_, k_] := Sum[k^(Floor[n/j] - 1), {j, 1, n}]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 06 2021 *)

Prog

(PARI) T(n, k) = sum(j=1, n, k^(n\j-1));

Crossrefs

Columns k=0..3 give A110654, A000027, A345028, A345029.

T(n,n) gives A345030.

Cf. A344821, A345033.

Sequence in context: A114115 A126268 A193739 * A182535 A181186 A364197

Adjacent sequences: A345029 A345030 A345031 * A345033 A345034 A345035

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 06 2021

Status

approved