A294810 -id:A294810 - cp's OEIS frontend

A294809 Expansion of Product_{k>=1} (1 - k^k*x^k)^k.

1, -1, -8, -73, -927, -13969, -254580, -5288596, -124795126, -3272571133, -94692028369, -2991756529687, -102571647087930, -3791499758414848, -150359326161180392, -6367668575791613601, -286854342016830115157, -13697147209040205869792

Offset: 0

Seiichi Manyama, Nov 09 2017

sign

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

Column k=1 of A294808.

Cf. A294810.

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} A294810(k)*a(n-k) for n > 0.

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

Original entry on oeis.org

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.

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

A294955 a(n) = Sum_{d|n} d^(2*n+2).

Original entry on oeis.org

1, 65, 6562, 1049601, 244140626, 78368963450, 33232930569602, 18014467229220865, 12157665462543713203, 10000002384185795209930, 9849732675807611094711842, 11447546167874515876354097130, 15502932802662396215269535105522

Offset: 1

Seiichi Manyama, Nov 12 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..213

Cf. A294645, A294810, A294953, A294954.

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

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

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

A308763 a(n) = Sum_{d|n} d^(n-2).

Original entry on oeis.org

1, 2, 4, 21, 126, 1394, 16808, 266305, 4785157, 100390882, 2357947692, 61978939050, 1792160394038, 56707753666594, 1946196290656824, 72061992352890881, 2862423051509815794, 121441386937936123331, 5480386857784802185940, 262145000003883417004506

Offset: 1

Seiichi Manyama, Jun 23 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A023887, A082245, A294645, A294810, A308755.

a[n_] := DivisorSum[n, #^(n - 2) &]; Array[a, 20] (* Amiram Eldar, May 08 2021 *)

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

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, (1-(k*x)^k)^(1/k^3)))))

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

L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)^(1/k^3)) = Sum_{k>=1} a(k)*x^k/k.

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

cp's OEIS Frontend

A294809 Expansion of Product_{k>=1} (1 - k^k*x^k)^k.

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

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A294955 a(n) = Sum_{d|n} d^(2*n+2).

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A308763 a(n) = Sum_{d|n} d^(n-2).

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