A344726 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/j)^k.

1, 1, 1, 1, 3, 3, 1, 7, 9, 2, 1, 15, 27, 12, 4, 1, 31, 81, 56, 22, 4, 1, 63, 243, 240, 118, 30, 6, 1, 127, 729, 992, 610, 196, 44, 4, 1, 255, 2187, 4032, 3094, 1230, 324, 48, 7, 1, 511, 6561, 16256, 15562, 7564, 2336, 448, 71, 7, 1, 1023, 19683, 65280, 77998, 45990, 16596, 3840, 685, 83, 9

Offset

1,5

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).

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

Example

Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  3,   7,   15,   31,    63, ...
  3,  9,  27,   81,  243,   729, ...
  2, 12,  56,  240,  992,  4032, ...
  4, 22, 118,  610, 3094, 15562, ...
  4, 30, 196, 1230, 7564, 45990, ...

Mathematica

T[n_, k_] := Sum[(-1)^(j + 1) * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 27 2021 *)

Prog

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

Crossrefs

Columns k=1..5 give A059851, A344720, A344721, A344722, A344723.

T(n,n) gives A344724.

Cf. A344725.

Sequence in context: A306759 A214362 A180735 * A116401 A106479 A114422

Adjacent sequences: A344723 A344724 A344725 * A344727 A344728 A344729

Keyword

nonn,tabl

Author

Seiichi Manyama, May 27 2021

Status

approved