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A322263
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = numerator of Sum_{d|n} 1/d^k.
1
1, 1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 10, 7, 2, 1, 17, 28, 21, 6, 4, 1, 33, 82, 73, 26, 2, 2, 1, 65, 244, 273, 126, 25, 8, 4, 1, 129, 730, 1057, 626, 7, 50, 15, 3, 1, 257, 2188, 4161, 3126, 697, 344, 85, 13, 4, 1, 513, 6562, 16513, 15626, 671, 2402, 585, 91, 9, 2, 1, 1025, 19684, 65793, 78126, 23725, 16808, 4369, 757, 13, 12, 6
OFFSET
1,3
FORMULA
G.f. of column k: Sum_{j>=1} x^j/(j^k*(1 - x^j)) (for rationals Sum_{d|n} 1/d^k).
Dirichlet g.f. of column k: zeta(s)*zeta(s+k) (for rationals Sum_{d|n} 1/d^k).
A(n,k) = numerator of sigma_k(n)/n^k.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
2, 3/2, 5/4, 9/8, 17/16, 33/32, ...
2, 4/3, 10/9, 28/27, 82/81, 244/243, ...
3, 7/4, 21/16, 73/64, 273/256, 1057/1024, ...
2, 6/5, 26/25, 126/125, 626/625, 3126/3125, ...
4, 2, 25/18, 7/6, 697/648, 671/648, ...
MATHEMATICA
Table[Function[k, Numerator[DivisorSigma[-k, n]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Numerator[DivisorSigma[k, n]/n^k]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, Numerator[SeriesCoefficient[Sum[x^j/(j^k (1 - x^j)), {j, 1, n}], {x, 0, n}]]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
KEYWORD
nonn,tabl,frac
AUTHOR
Ilya Gutkovskiy, Dec 01 2018
STATUS
approved