A294606 -id:A294606 - cp's OEIS frontend

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-jx^j)^(j^(kj)) in powers of x.

1, 1, -1, 1, -1, -2, 1, -1, -8, -1, 1, -1, -32, -73, -1, 1, -1, -128, -2155, -919, 5, 1, -1, -512, -58921, -259477, -13977, 1, 1, -1, -2048, -1593811, -67041751, -48496477, -253640, 13, 1, -1, -8192, -43044673, -17178144301, -152513231553, -13001163543, -5290184, 4

Offset: 0

Seiichi Manyama, Nov 04 2017

			Square array begins:
    1,    1,       1,         1,            1, ...
   -1,   -1,      -1,        -1,           -1, ...
   -2,   -8,     -32,      -128,         -512, ...
   -1,  -73,   -2155,    -58921,     -1593811, ...
   -1, -919, -259477, -67041751, -17178144301, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..2 give A022661, A294606, A294607.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A283675, A294609.

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k*d+1+j/d)) * A(n-j,k) for n > 0.

A294608 a(n) = Sum_{d|n} d^(d + 1 + n/d).

Original entry on oeis.org

1, 17, 244, 4129, 78126, 1680410, 40353608, 1073758337, 31381061797, 1000000390882, 34522712143932, 1283918474699170, 51185893014090758, 2177953338091847410, 98526125335695332184, 4722366482878235412481, 239072435685151324847154

Offset: 1

Seiichi Manyama, Nov 04 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..384

Cf. A294606.

a[n_] := DivisorSum[n, #^(# + 1 + n/#) &]; Array[a, 17] (* Amiram Eldar, Oct 04 2023 *)

PARI
```
{a(n) = sumdiv(n, d, d^(d+1+n/d))}
```

A294610 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^k).

Original entry on oeis.org

1, 1, 9, 90, 1154, 17427, 309117, 6285102, 144603015, 3717580810, 105696353842, 3293810230381, 111651093529948, 4089889271054734, 160989247361249558, 6776381334102511286, 303712681809603918633, 14439887378431671417669

Offset: 0

Seiichi Manyama, Nov 04 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..385

Column k=1 of A294609.

Cf. A294606, A294608.

nmax = 20; CoefficientList[Series[Product[1/(1 - k*x^k)^(k^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 05 2017 *)

N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-k*x^k)^k^k))

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294608(k) * a(n-k) for n > 0.

cp's OEIS Frontend

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-jx^j)^(j^(kj)) in powers of x.

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Formula

A294608 a(n) = Sum_{d|n} d^(d + 1 + n/d).

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Author

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Mathematica

PARI

A294610 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j*x^j)^(j^(k*j)) in powers of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A294608 a(n) = Sum_{d|n} d^(d + 1 + n/d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A294610 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-jx^j)^(j^(kj)) in powers of x.