A294809 -id:A294809 - cp's OEIS frontend

A294810 a(n) = Sum_{d|n} d^(n+2).

1, 17, 244, 4161, 78126, 1686434, 40353608, 1074791425, 31381236757, 1000244144722, 34522712143932, 1283997101947770, 51185893014090758, 2177986570740006274, 98526126098761952664, 4722384497336874434561, 239072435685151324847154

Offset: 1

Seiichi Manyama, Nov 09 2017

nonn

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

Cf. A294645, A294809.

Table[Total[Divisors[n]^(n+2)],{n,20}] (* Harvey P. Dale, Dec 23 2023 *)

PARI
```
{a(n) = sigma(n, n+2)}
```

N=66; x='x+O('x^N); Vec(sum(k=1, N, k^(k+2)*x^k/(1-(k*x)^k)))

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k*x)^k)^k)))) \\ Seiichi Manyama, Jun 02 2019

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

L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)^k) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 02 2019

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -8, -1, 1, -1, -32, -73, 0, 1, -1, -128, -2155, -927, 4, 1, -1, -512, -58921, -259701, -13969, 4, 1, -1, -2048, -1593811, -67045719, -48496253, -254580, 7, 1, -1, -8192, -43044673, -17178209325, -152513227585, -13001952944, -5288596, 3

Offset: 0

Seiichi Manyama, Nov 09 2017

			Square array begins:
    1,      1,         1,             1,                1, ...
   -1,     -1,        -1,            -1,               -1, ...
   -2,     -8,       -32,          -128,             -512, ...
   -1,    -73,     -2155,        -58921,         -1593811, ...
    0,   -927,   -259701,     -67045719,     -17178209325, ...
    4, -13969, -48496253, -152513227585, -476819162106101, ...

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

Columns k=0..2 give A073592, A294809, A294953.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A004171.
Cf. A294653, A294950.

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j)) * A(n-j,k) for n > 0.

A294813 Expansion of Product_{k>=1} 1/(1 - k^k*x^k)^k.

Original entry on oeis.org

1, 1, 9, 90, 1162, 17435, 310193, 6286826, 144750451, 3717959194, 105725550762, 3293914191401, 111659484775650, 4089936343858976, 160992739588472076, 6776415674628574634, 303714862444753023205, 14439925495117621425535

Offset: 0

Seiichi Manyama, Nov 09 2017

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n, g(n) = n^n.

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

Cf. A294809, A294810.

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

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A294810(k)*a(n-k) for n > 0.

a(n) ~ n^(n+1). - Vaclav Kotesovec, Nov 10 2017

A294947 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(-Sum_{j>0} sigma_k(j)*x^j/j) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -4, -7, 1, -1, -8, -23, -57, 1, -1, -16, -73, -229, -541, 1, -1, -32, -227, -927, -2761, -7126, 1, -1, -64, -697, -3757, -13969, -42615, -108072, 1, -1, -128, -2123, -15207, -70237, -254580, -758499, -1966034

Offset: 0

Seiichi Manyama, Nov 11 2017

			Square array begins:
      1,     1,      1,      1,       1, ...
     -1,    -1,     -1,     -1,      -1, ...
     -2,    -4,     -8,    -16,     -32, ...
     -7,   -23,    -73,   -227,    -697, ...
    -57,  -229,   -927,  -3757,  -15207, ...
   -541, -2761, -13969, -70237, -351361, ...

Columns k=0..2 give A294948, A292312, A294809.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A294946, A294951.

G.f. of column k: Product_{j>0} (1 - j^j*x^j)^(j^(k-1)).

A295244 Expansion of Product_{k>=1} (1 + k^k*x^k)^k.

Original entry on oeis.org

1, 1, 8, 89, 1121, 17313, 306588, 6264356, 144123978, 3710660797, 105517536463, 3289885612007, 111534718552758, 4086488012360688, 160872398645645560, 6772203940050042913, 303547711924399057195, 14433006933542826499824

Offset: 0

Seiichi Manyama, Nov 18 2017

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n, g(n) = -n^n.

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

Cf. A266964, A294809, A295245.

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

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^(2+n)*(-1)^(1+n/d).

cp's OEIS Frontend

A294810 a(n) = Sum_{d|n} d^(n+2).

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A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(k*j)*x^j)^j in powers of x.

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A294813 Expansion of Product_{k>=1} 1/(1 - k^k*x^k)^k.

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A294947 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(-Sum_{j>0} sigma_k(j)*x^j/j) in powers of x.

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A295244 Expansion of Product_{k>=1} (1 + k^k*x^k)^k.

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Formula

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.