A320973 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = n^k * Product_{p|n, p prime} (1 + 1/p^k).

1, 1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 9, 10, 6, 2, 1, 17, 28, 20, 6, 4, 1, 33, 82, 72, 26, 12, 2, 1, 65, 244, 272, 126, 50, 8, 2, 1, 129, 730, 1056, 626, 252, 50, 12, 2, 1, 257, 2188, 4160, 3126, 1394, 344, 80, 12, 4, 1, 513, 6562, 16512, 15626, 8052, 2402, 576, 90, 18, 2

Offset

1,3

Links

Table of n, a(n) for n=1..66.

Wikipedia, Dedekind psi function

Formula

G.f. of column k: Sum_{j>=1} mu(j)^2*PolyLog(-k,x^j), where PolyLog() is the polylogarithm function.

A(n,k) = Sum_{d|n} mu(n/d)^2*d^k.

Example

Square array begins:
  1,   1,   1,    1,     1,     1,  ...
  2,   3,   5,    9,    17,    33,  ...
  2,   4,  10,   28,    82,   244,  ...
  2,   6,  20,   72,   272,  1056,  ...
  2,   6,  26,  126,   626,  3126,  ...
  4,  12,  50,  252,  1394,  8052,  ...

Mathematica

Table[Function[k, n^k Product[1 + Boole[PrimeQ[d]]/d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[MoebiusMu[j]^2 PolyLog[-k, x^j], {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[MoebiusMu[n/d]^2 d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

Crossrefs

Columns k=0..4 give A034444, A001615, A065958, A065959, A065960.

Cf. A008683, A059379, A059380, A320974 (diagonal).

Sequence in context: A058399 A209434 A207611 * A058400 A131344 A129262

Adjacent sequences: A320970 A320971 A320972 * A320974 A320975 A320976

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Oct 25 2018

Status

approved