A344824 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j) * (-k)^(j-1).

1, 1, 2, 1, 1, 3, 1, 0, 3, 4, 1, -1, 5, 2, 5, 1, -2, 9, -4, 4, 6, 1, -3, 15, -20, 13, 4, 7, 1, -4, 23, -52, 62, -16, 6, 8, 1, -5, 33, -106, 205, -174, 49, 4, 9, 1, -6, 45, -188, 520, -806, 556, -88, 7, 10, 1, -7, 59, -304, 1109, -2584, 3291, -1660, 173, 7, 11

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} x^j/(1 + k*x^j).

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

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

Example

Square array begins:
  1, 1,   1,    1,    1,     1,     1, ...
  2, 1,   0,   -1,   -2,    -3,    -4, ...
  3, 3,   5,    9,   15,    23,    33, ...
  4, 2,  -4,  -20,  -52,  -106,  -188, ...
  5, 4,  13,   62,  205,   520,  1109, ...
  6, 4, -16, -174, -806, -2584, -6636, ...

Mathematica

T[n_, k_] := Sum[If[k == 0 && j == 1, 1, (-k)^(j - 1)] * Quotient[n, j], {j, 1, n}]; Table[T[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 29 2021 *)

Prog

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

Crossrefs

Columns k=0..4 give A000027, A059851, A344817, A344818, A344819.

T(n,n) gives A344820.

Cf. A344821.

Sequence in context: A227431 A114118 A146014 * A226173 A212633 A202241

Adjacent sequences: A344821 A344822 A344823 * A344825 A344826 A344827

Keyword

sign,tabl

Author

Seiichi Manyama, May 29 2021

Status

approved