A350106 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j * floor(n/j)^k.

1, 1, 4, 1, 6, 8, 1, 10, 14, 15, 1, 18, 32, 31, 21, 1, 34, 86, 87, 45, 33, 1, 66, 248, 295, 153, 81, 41, 1, 130, 734, 1095, 669, 309, 101, 56, 1, 258, 2192, 4231, 3201, 1521, 443, 150, 69, 1, 514, 6566, 16647, 15765, 8373, 2633, 722, 191, 87, 1, 1026, 19688, 66055, 78393, 48321, 17411, 4746, 1005, 253, 99

Offset

1,3

Links

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Formula

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^j)^2.

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

Example

Square array begins:
   1,   1,   1,    1,     1,      1,      1, ...
   4,   6,  10,   18,    34,     66,    130, ...
   8,  14,  32,   86,   248,    734,   2192, ...
  15,  31,  87,  295,  1095,   4231,  16647, ...
  21,  45, 153,  669,  3201,  15765,  78393, ...
  33,  81, 309, 1521,  8373,  48321, 284709, ...
  41, 101, 443, 2633, 17411, 119321, 828323, ...

Mathematica

T[n_, k_] := Sum[j * Floor[n/j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 14 2021 *)

Prog

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

Crossrefs

Columns k=1..3 give A024916, A350107, A350108.

T(n,n) gives A350109.

Cf. A319649, A344725.

Sequence in context: A373918 A134234 A207628 * A205137 A223534 A222391

Adjacent sequences: A350103 A350104 A350105 * A350107 A350108 A350109

Keyword

nonn,tabl

Author

Seiichi Manyama, Dec 14 2021

Status

approved