A308504 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d^(n+k).

1, 1, 5, 1, 9, 28, 1, 17, 82, 273, 1, 33, 244, 1057, 3126, 1, 65, 730, 4161, 15626, 47450, 1, 129, 2188, 16513, 78126, 282252, 823544, 1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 1, 513, 19684, 262657, 1953126, 10097892, 40353608, 134480385, 387440173

Offset

1,3

Links

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Index entries for sequences related to sigma(n)

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - (j*x)^j)^(j^(k-1))).

a((i-1)*i/2 + j) = sigma_i(j) for 1 <= j <= i.

Example

a(4) = a(2*3/2 + 1) = sigma_3(1) = 1.
a(5) = a(2*3/2 + 2) = sigma_3(2) = 1^3 + 2^3 = 9.
a(6) = a(2*3/2 + 3) = sigma_3(3) = 1^3 + 3^3 = 28.
Square array begins:
       1,      1,       1,        1,        1, ...
       5,      9,      17,       33,       65, ...
      28,     82,     244,      730,     2188, ...
     273,   1057,    4161,    16513,    65793, ...
    3126,  15626,   78126,   390626,  1953126, ...
   47450, 282252, 1686434, 10097892, 60526250, ...

Mathematica

T[n_, k_] := DivisorSum[n, #^(n+k) &]; Table[T[k, n - k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, May 11 2021 *)

Crossrefs

Columns k=0..2 give A023887, A294645, A294810.

A(n,n) gives A308570.

Cf. A279394, A308502.

Sequence in context: A147404 A254068 A146070 * A040029 A105086 A305838

Adjacent sequences: A308501 A308502 A308503 * A308505 A308506 A308507

Keyword

nonn,tabl

Author

Seiichi Manyama, Jun 02 2019

Status

approved