A345033 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-k)^(floor(n/j) - 1).

1, 1, 1, 1, 0, 2, 1, -1, 3, 2, 1, -2, 6, 0, 3, 1, -3, 11, -8, 3, 3, 1, -4, 18, -28, 17, 2, 4, 1, -5, 27, -66, 81, -27, 5, 4, 1, -6, 38, -128, 255, -234, 70, 0, 5, 1, -7, 51, -220, 623, -1009, 739, -136, 5, 5, 1, -8, 66, -348, 1293, -3102, 4112, -2216, 255, 4, 6, 1, -9, 83, -518, 2397, -7743, 15649, -16452, 6545, -491, 7, 6

Offset

1,6

Links

Table of n, a(n) for n=1..78.

Formula

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

Example

Square array begins:
  1, 1,   1,    1,     1,     1,     1, ...
  1, 0,  -1,   -2,    -3,    -4,    -5, ...
  2, 3,   6,   11,    18,    27,    38, ...
  2, 0,  -8,  -28,   -66,  -128,  -220, ...
  3, 3,  17,   81,   255,   623,  1293, ...
  3, 2, -27, -234, -1009, -3102, -7743, ...

Mathematica

T[n_, 0] := Floor[(n + 1)/2]; T[n_, k_] := Sum[(-k)^(Floor[n/j] - 1), {j, 1, n}]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 06 2021 *)

Prog

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

Crossrefs

Columns k=0..3 give A110654, A271860, A345034, A345035.

T(n,n) gives A345036.

Cf. A344824, A345032.

Sequence in context: A273104 A367680 A152538 * A329359 A235682 A324830

Adjacent sequences: A345030 A345031 A345032 * A345034 A345035 A345036

Keyword

sign,tabl

Author

Seiichi Manyama, Jun 06 2021

Status

approved