A356250 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (j * floor(n/j))^k.

1, 1, 2, 1, 4, 3, 1, 8, 8, 4, 1, 16, 22, 15, 5, 1, 32, 62, 57, 21, 6, 1, 64, 178, 219, 91, 33, 7, 1, 128, 518, 849, 405, 185, 41, 8, 1, 256, 1522, 3315, 1843, 1053, 247, 56, 9, 1, 512, 4502, 13017, 8541, 6065, 1523, 402, 69, 10, 1, 1024, 13378, 51339, 40171, 35253, 9571, 2948, 545, 87, 11

Offset

1,3

Links

Table of n, a(n) for n=1..66.

Formula

T(n,k) = Sum_{j=1..n} j^k * Sum_{d|j} (1 - (1 - 1/d)^k) for k > 0.

Example

Square array begins:
  1,  1,   1,    1,    1,     1,      1, ...
  2,  4,   8,   16,   32,    64,    128, ...
  3,  8,  22,   62,  178,   518,   1522, ...
  4, 15,  57,  219,  849,  3315,  13017, ...
  5, 21,  91,  405, 1843,  8541,  40171, ...
  6, 33, 185, 1053, 6065, 35253, 206345, ...
  7, 41, 247, 1523, 9571, 61091, 394987, ...

Mathematica

T[n_, k_] := Sum[(j * Floor[n/j])^k, {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 31 2022 *)

Prog

(PARI) T(n, k) = sum(j=1, n, (j*(n\j))^k);
(PARI) T(n, k) = if(k==0, n, sum(j=1, n, j^k*sumdiv(j, d, 1-(1-1/d)^k)));

Crossrefs

Columns k=0..3 give A001477, A024916, A350123, A356249.

T(n,n) gives A356238.

Cf. A344725.

Sequence in context: A105851 A106195 A247023 * A348702 A051129 A319075

Adjacent sequences: A356247 A356248 A356249 * A356251 A356252 A356253

Keyword

nonn,tabl

Author

Seiichi Manyama, Jul 31 2022

Status

approved