A350122 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/(2*j-1))^k.

1, 1, 2, 1, 4, 4, 1, 8, 10, 5, 1, 16, 28, 17, 7, 1, 32, 82, 65, 27, 9, 1, 64, 244, 257, 127, 41, 11, 1, 128, 730, 1025, 627, 225, 55, 12, 1, 256, 2188, 4097, 3127, 1313, 353, 70, 15, 1, 512, 6562, 16385, 15627, 7809, 2419, 522, 93, 17, 1, 1024, 19684, 65537, 78127, 46721, 16841, 4114, 759, 115, 19

Offset

1,3

Links

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

Formula

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

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

Example

Square array begins:
   1,  1,   1,    1,     1,      1,      1, ...
   2,  4,   8,   16,    32,     64,    128, ...
   4, 10,  28,   82,   244,    730,   2188, ...
   5, 17,  65,  257,  1025,   4097,  16385, ...
   7, 27, 127,  627,  3127,  15627,  78127, ...
   9, 41, 225, 1313,  7809,  46721, 280065, ...
  11, 55, 353, 2419, 16841, 117715, 823673, ...

Mathematica

T[n_, k_] := Sum[Floor[n/(2*j - 1)]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 17 2021 *)

Prog

(PARI) T(n, k) = sum(j=1, n, (n\(2*j-1))^k);
(PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, j/d%2*(d^k-(d-1)^k)));

Crossrefs

Columns k=1..3 give A060831, A350143, A350144.

T(n,n) gives A350145.

Cf. A344725.

Sequence in context: A220537 A229717 A122438 * A156708 A131250 A140693

Adjacent sequences: A350119 A350120 A350121 * A350123 A350124 A350125

Keyword

nonn,tabl

Author

Seiichi Manyama, Dec 16 2021

Status

approved