A308570 -id:A308570 - cp's OEIS frontend

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

Seiichi Manyama, Jun 02 2019

			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, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Index entries for sequences related to sigma(n)

Columns k=0..2 give A023887, A294645, A294810.
A(n,n) gives A308570.
Cf. A279394, A308502.

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

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.

A308569 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n).

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 1, 17, 28, 3, 1, 65, 730, 273, 2, 1, 257, 19684, 65793, 3126, 4, 1, 1025, 531442, 16781313, 9765626, 47450, 2, 1, 4097, 14348908, 4295032833, 30517578126, 2177317874, 823544, 4, 1, 16385, 387420490, 1099512676353, 95367431640626, 101560344351050, 678223072850, 16843009, 3

Offset: 1

Seiichi Manyama, Jun 08 2019

			Square array begins:
   1,    1,       1,           1,              1, ...
   2,    5,      17,          65,            257, ...
   2,   28,     730,       19684,         531442, ...
   3,  273,   65793,    16781313,     4295032833, ...
   2, 3126, 9765626, 30517578126, 95367431640626, ...

Seiichi Manyama, Antidiagonals n = 1..52, flattened

Columns k=0..2 give A000005, A023887, A308570.
Rows n=1..2 give A000012, A052539.
A(n,n) gives A308571.
Cf. A308504, A308509.

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

T(n,k) = sumdiv(n, d, d^(k*n));
matrix(5, 5, n, k, T(n,k-1)) \\ Michel Marcus, Jun 08 2019

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

A367551 a(n) = Sum_{d|n} (d^2+1)^n.

Original entry on oeis.org

2, 29, 1008, 84162, 11881408, 2566742098, 781250000128, 318651789038947, 167619551409708544, 110462353708225871026, 89116503268220597579776, 86380568889558343409300388, 99045780329059370000000008192

Offset: 1

Seiichi Manyama, Nov 22 2023

nonn

Cf. A163190, A308570, A367552.

PARI
```
a(n) = sumdiv(n, d, (d^2+1)^n);
```

a(n) = Sum_{k=0..n} binomial(n,k) * sigma_{2*k}(n).

cp's OEIS Frontend

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

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A308569 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(k*n).

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Examples

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Programs

Mathematica

PARI

Formula

A367551 a(n) = Sum_{d|n} (d^2+1)^n.

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PARI

Formula