OFFSET
1,3
COMMENTS
For each k, the k-th column sequence (T(n,k))(n>=1) is a multiplicative function of n, equal to (-1)^(n+1)*(Id_k * 1) in the notation of the Bala link. - Peter Bala, Mar 19 2022
LINKS
FORMULA
G.f. of column k: Sum_{j>=1} (-1)^(j+1)*j^k*x^j/(1 + x^j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
-2, -3, -5, -9, -17, -33, ...
2, 4, 10, 28, 82, 244, ...
-1, -3, -13, -57, -241, -993, ...
2, 6, 26, 126, 626, 3126, ...
-4, -12, -50, -252, -1394, -8052, ...
MATHEMATICA
Table[Function[k, Sum[(-1)^(n/d+d) d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[(-1)^(j + 1) j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
f[p_, e_, k_] := If[k == 0, e + 1, (p^(k*e + k) - 1)/(p^k - 1)]; f[2, e_, k_] := If[k == 0, e - 3, -((2^(k - 1) - 1)*2^(k*e + 1) + 2^(k + 1) - 1)/(2^k - 1)]; T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k, k], {n, 1, 11}, {k, n - 1, 0, -1}] // Flatten (* Amiram Eldar, Nov 22 2022 *)
PROG
(PARI) T(n, k)={sumdiv(n, d, (-1)^(n/d+d)*d^k)}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Ilya Gutkovskiy, Nov 26 2018
STATUS
editing