A349883 -id:A349883 - cp's OEIS frontend

A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

1, 1, 5, 44, 564, 9665, 211025, 5686104, 184813048, 7118824417, 320295658577, 16626717667348, 985178854556524, 66005199079345025, 4958773228726876257, 414664315430994701616, 38344259607889223269168, 3898112616839310343827009

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..299

Cf. A031971, A349883.

Cf. A249459, A349884, A368505.

Join[{1}, Table[Sum[k^(2*n - k), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, t=2) = sum(k=0, n, k^(t*(n-k)+k));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^2*x)))

a(n) = Sum_{k=0..n} k^(2*n-k).

a(n) ~ sqrt(Pi) * 2^(1 + 2*n - 2*n/LambertW(2*exp(1)*n)) * (n/LambertW(2*exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(2*exp(1)*n)) / sqrt(1 + LambertW(2*exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A349901 a(n) = Sum_{k=0..n} k^(3*n).

Original entry on oeis.org

1, 1, 65, 20196, 17312754, 31605701625, 105443761093411, 580964060390826448, 4918745981990731659972, 60634331963604550954204425, 1043651859661187698792930519525, 24256699178432730349549665042311076, 740737411098120942914045235001015624310

Offset: 0

Seiichi Manyama, Dec 05 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..152

Cf. A249459, A349883, A349886.

PARI
```
a(n) = sum(k=0, n, k^(3*n));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^3*x)^k/(1-k^3*x)))

G.f.: Sum_{k>=0} (k^3 * x)^k/(1 - k^3 * x).

a(n) ~ exp(3)/(exp(3)-1) * n^(3*n). - Vaclav Kotesovec, Dec 05 2021

A359659 a(n) = Sum_{k=0..n} k^(k * (n-k+1)).

Original entry on oeis.org

1, 2, 6, 45, 1051, 88602, 27121964, 37004504305, 198705527223757, 5595513387083114570, 686714367475480207331582, 468422339816915120237104999421, 1664212116512828935888786624225704855

Offset: 0

Seiichi Manyama, Jan 10 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..51

Cf. A026898, A349893, A359658.
Cf. A031971, A349836, A349883.
Cf. A003101, A349882.

PARI
```
a(n) = sum(k=0, n, k^(k*(n-k+1)));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^k*x)))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k+1)^(k+1)*x)))

G.f.: Sum_{k>=0} (k * x)^k/(1 - k^k * x).
G.f.: Sum_{k>=0} x^k/(1 - (k+1)^(k+1) * x).
a(n) = A349893(n+1) - 1.

cp's OEIS Frontend

A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

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A349901 a(n) = Sum_{k=0..n} k^(3*n).

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A359659 a(n) = Sum_{k=0..n} k^(k * (n-k+1)).

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