A308688 -id:A308688 - cp's OEIS frontend

A308690 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*n/d - k + 1), read by antidiagonals.

1, 1, 3, 1, 3, 4, 1, 3, 4, 7, 1, 3, 4, 9, 6, 1, 3, 4, 13, 6, 12, 1, 3, 4, 21, 6, 24, 8, 1, 3, 4, 37, 6, 66, 8, 15, 1, 3, 4, 69, 6, 216, 8, 41, 13, 1, 3, 4, 133, 6, 762, 8, 201, 37, 18, 1, 3, 4, 261, 6, 2784, 8, 1289, 253, 68, 12, 1, 3, 4, 517, 6, 10386, 8, 9225, 2197, 648, 12, 28

Offset: 1

Seiichi Manyama, Jun 17 2019

			Square array begins:
    1,  1,  1,   1,   1,    1,     1, ...
    3,  3,  3,   3,   3,    3,     3, ...
    4,  4,  4,   4,   4,    4,     4, ...
    7,  9, 13,  21,  37,   69,   133, ...
    6,  6,  6,   6,   6,    6,     6, ...
   12, 24, 66, 216, 762, 2784, 10386, ...
    8,  8,  8,   8,   8,    8,     8, ...

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

Columns k=0..3 give A000203, A055225, A308688, A308689.

Cf. A294579, A308509, A308694.

T[n_, k_] := DivisorSum[n, #^(k*n/# - k + 1) &]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

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

A(p,k) = p+1 for prime p.

A356560 Expansion of e.g.f. Product_{k>0} 1/(1 - k^2 * x^k)^(1/k^2).

Original entry on oeis.org

1, 1, 4, 18, 156, 1020, 16560, 143640, 2898000, 43016400, 926856000, 13749674400, 524416939200, 8626888670400, 284030505158400, 7950850859952000, 284397434953632000, 6752059834744224000, 357295791069689472000, 9098085523917918528000

Offset: 0

Seiichi Manyama, Aug 12 2022

nonn

Cf. A294462, A294469, A356530, A356561.

Cf. A077335, A308688.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k^2*x^k)^(1/k^2))))

a308688(n) = sumdiv(n, d, d^(2*n/d-1));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, a308688(j)*v[i-j+1]/(i-j)!)); v;

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A308688(k) * a(n-k)/(n-k)!.

A359732 a(n) = Sum_{d|n} d^(2*d-1).

Original entry on oeis.org

1, 9, 244, 16393, 1953126, 362797308, 96889010408, 35184372105225, 16677181699666813, 10000000000001953134, 7400249944258160101212, 6624737266949237373933820, 7056410014866816666030739694, 8819763977946281130541873428720

Offset: 1

Seiichi Manyama, Jan 12 2023

Cf. A308688, A308696, A359731.

a[n_] := DivisorSum[n, #^(2*# - 1) &]; Array[a, 15] (* Amiram Eldar, Aug 14 2023 *)

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

my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, k^(2*k-1)*x^k/(1-x^k)))

G.f.: Sum_{k>0} k^(2*k-1) * x^k / (1 - x^k).

cp's OEIS Frontend

A308690 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*n/d - k + 1), read by antidiagonals.

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A356560 Expansion of e.g.f. Product_{k>0} 1/(1 - k^2 * x^k)^(1/k^2).

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PARI

Formula

A359732 a(n) = Sum_{d|n} d^(2*d-1).

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