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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).
6
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
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.
Sequence in context: A273104 A367680 A152538 * A329359 A235682 A324830
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Jun 06 2021
STATUS
approved