A350161 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/(2*j-1))^k.

1, 1, 2, 1, 4, 2, 1, 8, 8, 3, 1, 16, 26, 15, 5, 1, 32, 80, 63, 25, 5, 1, 64, 242, 255, 125, 33, 5, 1, 128, 728, 1023, 625, 209, 45, 6, 1, 256, 2186, 4095, 3125, 1281, 335, 60, 7, 1, 512, 6560, 16383, 15625, 7745, 2385, 504, 73, 9, 1, 1024, 19682, 65535, 78125, 46593, 16775, 4080, 703, 95, 9

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} A101455(j/d) * (d^k - (d - 1)^k).

Example

Square array begins:
  1,  1,   1,    1,     1,      1,      1, ...
  2,  4,   8,   16,    32,     64,    128, ...
  2,  8,  26,   80,   242,    728,   2186, ...
  3, 15,  63,  255,  1023,   4095,  16383, ...
  5, 25, 125,  625,  3125,  15625,  78125, ...
  5, 33, 209, 1281,  7745,  46593, 279809, ...
  5, 45, 335, 2385, 16775, 117585, 823415, ...

Mathematica

T[n_, k_] := Sum[(-1)^(j + 1) * 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 18 2021 *)

Prog

(PARI) T(n, k) = sum(j=1, n, (-1)^(j+1)*(n\(2*j-1))^k);
(PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, kronecker(-4, j/d)*(d^k-(d-1)^k)));

Crossrefs

Columns k=1..3 give A014200, A350162, A350163.

T(n,n) gives A350164.

Cf. A101455, A344726, A350122.

Sequence in context: A157028 A060637 A123486 * A158264 A274106 A354802

Adjacent sequences: A350158 A350159 A350160 * A350162 A350163 A350164

Keyword

nonn,tabl

Author

Seiichi Manyama, Dec 18 2021

Status

approved