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A350161
Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-1)^(j+1) * floor(n/(2*j-1))^k.
3
1, 1, 2, 1, 4, 2, 1, 8, 8, 3, 1, 16, 26, 15, 5, 1, 32, 80, 63, 25, 5, 1, 64, 242, 255, 125, 33, 5, 1, 128, 728, 1023, 625, 209, 45, 6, 1, 256, 2186, 4095, 3125, 1281, 335, 60, 7, 1, 512, 6560, 16383, 15625, 7745, 2385, 504, 73, 9, 1, 1024, 19682, 65535, 78125, 46593, 16775, 4080, 703, 95, 9
OFFSET
1,3
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} A101455(j/d) * (d^k - (d - 1)^k).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
2, 4, 8, 16, 32, 64, 128, ...
2, 8, 26, 80, 242, 728, 2186, ...
3, 15, 63, 255, 1023, 4095, 16383, ...
5, 25, 125, 625, 3125, 15625, 78125, ...
5, 33, 209, 1281, 7745, 46593, 279809, ...
5, 45, 335, 2385, 16775, 117585, 823415, ...
MATHEMATICA
T[n_, k_] := Sum[(-1)^(j + 1) * 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 18 2021 *)
PROG
(PARI) T(n, k) = sum(j=1, n, (-1)^(j+1)*(n\(2*j-1))^k);
(PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, kronecker(-4, j/d)*(d^k-(d-1)^k)));
CROSSREFS
Columns k=1..3 give A014200, A350162, A350163.
T(n,n) gives A350164.
Sequence in context: A157028 A060637 A123486 * A158264 A274106 A354802
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Dec 18 2021
STATUS
approved