A345032 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, 2, 2, 1, 3, 3, 2, 1, 4, 6, 4, 3, 1, 5, 11, 12, 5, 3, 1, 6, 18, 32, 21, 6, 4, 1, 7, 27, 70, 87, 41, 7, 4, 1, 8, 38, 132, 263, 258, 74, 8, 5, 1, 9, 51, 224, 633, 1047, 745, 144, 9, 5, 1, 10, 66, 352, 1305, 3158, 4120, 2224, 275, 10, 6, 1, 11, 83, 522, 2411, 7821, 15659, 16460, 6605, 541, 11, 6
Offset: 1
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, 6, 7, ... 2, 3, 6, 11, 18, 27, 38, ... 2, 4, 12, 32, 70, 132, 224, ... 3, 5, 21, 87, 263, 633, 1305, ... 3, 6, 41, 258, 1047, 3158, 7821, ...
Crossrefs
Programs
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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 *) -
PARI
T(n, k) = sum(j=1, n, k^(n\j-1));
Formula
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j * (1 - x^j)/(1 - k*x^j).