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A321258
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = sigma_k(n) - n^k.
5
0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 1, 0, 1, 1, 5, 1, 3, 0, 1, 1, 9, 1, 6, 1, 0, 1, 1, 17, 1, 14, 1, 3, 0, 1, 1, 33, 1, 36, 1, 7, 2, 0, 1, 1, 65, 1, 98, 1, 21, 4, 3, 0, 1, 1, 129, 1, 276, 1, 73, 10, 8, 1, 0, 1, 1, 257, 1, 794, 1, 273, 28, 30, 1, 5
OFFSET
1,10
COMMENTS
A(n,k) is the sum of k-th powers of proper divisors of n.
FORMULA
G.f. of column k: Sum_{j>=1} j^k*x^(2*j)/(1 - x^j).
Dirichlet g.f. of column k: zeta(s-k)*(zeta(s) - 1).
A(n,k) = 1 if n is prime.
EXAMPLE
Square array begins:
0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
2, 3, 5, 9, 17, 33, ...
1, 1, 1, 1, 1, 1, ...
3, 6, 14, 36, 98, 276, ...
MATHEMATICA
Table[Function[k, DivisorSigma[k, n] - n^k][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j^k x^(2 j)/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
CROSSREFS
Columns k=0..5 give A032741, A001065, A067558, A276634, A279363, A279364.
Cf. A109974, A285425, A286880, A321259 (diagonal).
Sequence in context: A217760 A339218 A263412 * A331510 A319854 A124035
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Nov 01 2018
STATUS
approved