login
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.
4
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
1,3
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, j/d odd} d^k - (d - 1)^k.
EXAMPLE
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, ...
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 *)
PROG
(PARI) T(n, k) = sum(j=1, n, (n\(2*j-1))^k);
(PARI) T(n, k) = sum(j=1, n, sumdiv(j, d, j/d%2*(d^k-(d-1)^k)));
CROSSREFS
Columns k=1..3 give A060831, A350143, A350144.
T(n,n) gives A350145.
Cf. A344725.
Sequence in context: A220537 A229717 A122438 * A156708 A131250 A140693
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Dec 16 2021
STATUS
approved