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A322103
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} sigma_k(d).
3
1, 1, 3, 1, 4, 3, 1, 6, 5, 6, 1, 10, 11, 11, 3, 1, 18, 29, 27, 7, 9, 1, 34, 83, 83, 27, 20, 3, 1, 66, 245, 291, 127, 66, 9, 10, 1, 130, 731, 1091, 627, 290, 51, 26, 6, 1, 258, 2189, 4227, 3127, 1494, 345, 112, 18, 9, 1, 514, 6563, 16643, 15627, 8330, 2403, 668, 102, 28, 3
OFFSET
1,3
LINKS
FORMULA
G.f. of column k: Sum_{j>=1} sigma_k(j)*x^j/(1 - x^j).
A(n,k) = Sum_{d|n} d^k*tau(n/d).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
3, 4, 6, 10, 18, 34, ...
3, 5, 11, 29, 83, 245, ...
6, 11, 27, 83, 291, 1091, ...
3, 7, 27, 127, 627, 3127, ...
9, 20, 66, 290, 1494, 8330, ...
MATHEMATICA
Table[Function[k, Sum[DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
PROG
(PARI) T(n, k)={sumdiv(n, d, d^k*numdiv(n/d))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018
CROSSREFS
Columns k=0..3 give A007425, A007429, A007433, A321140.
Cf. A109974, A321141 (diagonal), A356045.
Sequence in context: A104568 A030758 A347065 * A272172 A104764 A152842
KEYWORD
nonn,tabl
AUTHOR
Ilya Gutkovskiy, Nov 26 2018
STATUS
approved