A350122 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/(2*j-1))^k.
1, 1, 2, 1, 4, 4, 1, 8, 10, 5, 1, 16, 28, 17, 7, 1, 32, 82, 65, 27, 9, 1, 64, 244, 257, 127, 41, 11, 1, 128, 730, 1025, 627, 225, 55, 12, 1, 256, 2188, 4097, 3127, 1313, 353, 70, 15, 1, 512, 6562, 16385, 15627, 7809, 2419, 522, 93, 17, 1, 1024, 19684, 65537, 78127, 46721, 16841, 4114, 759, 115, 19
Offset: 1
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 2, 4, 8, 16, 32, 64, 128, ... 4, 10, 28, 82, 244, 730, 2188, ... 5, 17, 65, 257, 1025, 4097, 16385, ... 7, 27, 127, 627, 3127, 15627, 78127, ... 9, 41, 225, 1313, 7809, 46721, 280065, ... 11, 55, 353, 2419, 16841, 117715, 823673, ...
Programs
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Mathematica
T[n_, k_] := Sum[Floor[n/(2*j - 1)]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Dec 17 2021 *) -
PARI
T(n, k) = sum(j=1, n, (n\(2*j-1))^k);
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PARI
T(n, k) = sum(j=1, n, sumdiv(j, d, j/d%2*(d^k-(d-1)^k)));
Formula
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} (j^k - (j - 1)^k) * x^j/(1 - x^(2*j)).
T(n,k) = Sum_{j=1..n} Sum_{d|j, j/d odd} d^k - (d - 1)^k.