A349886 -id:A349886 - cp's OEIS frontend

A349893 a(n) = Sum_{k=0..n} k^(k*(n-k)).

1, 2, 3, 7, 46, 1052, 88603, 27121965, 37004504306, 198705527223758, 5595513387083114571, 686714367475480207331583, 468422339816915120237104999422, 1664212116512828935888786624225704856, 31295654819650678010096952493864470025103251

Offset: 0

Seiichi Manyama, Dec 04 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..52

Cf. A117402, A349880, A349881, A349886, A349894.

Table[1 + Sum[k^(k*(n - k)), {k, 1, n}], {n, 0, 16}] (* Vaclav Kotesovec, Dec 05 2021 *)

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

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

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

log(a(n)) ~ n^2*log(n)/4 * (1 - log(2)/log(n) + 1/(4*log(n)^2)). - Vaclav Kotesovec, Dec 05 2021

A349891 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(k*n).

Original entry on oeis.org

1, 0, 16, 19619, 4294436111, 298022124379673232, 10314423867168242405282727694, 256923577039829077600620024397823949901879, 6277101735175093150055816289268196664555481440709896684157

Offset: 0

Seiichi Manyama, Dec 04 2021

nonn

Cf. A120485, A349886, A349889.

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

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, k^k^2*x^k/(1+k^k*x)))

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

a(n) ~ n^(n^2). - Vaclav Kotesovec, Dec 10 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

A355466 Expansion of Sum_{k>=0} (k^k * x)^k/(1 - k^k * x)^(k+1).

Original entry on oeis.org

1, 2, 19, 19879, 4297094601, 298028721578591321, 10314430386430205371442173873, 256923580889667562995278943476559835493321, 6277101737079381674883855772624745947410338680458857322625

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A072034, A242446, A355470.

Cf. A349886.

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, (k^k*x)^k/(1-k^k*x)^(k+1)))

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(k^k*x)*(k^k*x)^k/k!)))

a(n) = sum(k=0, n, k^(k*n)*binomial(n, k));

E.g.f.: Sum_{k>=0} exp(k^k * x) * (k^k * x)^k/k!.

a(n) = Sum_{k=0..n} k^(k*n) * binomial(n,k).

A356689 a(n) = n! * Sum_{k=0..n} k^(k*n)/k!.

Original entry on oeis.org

1, 2, 20, 19887, 4297096180, 298028721722131825, 10314430386434427534836297166, 256923580889667624113335512704714686054849, 6277101737079381675512518990977258744796239498871290255000

Offset: 0

Seiichi Manyama, Aug 23 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A256016, A356687, A356688.

Cf. A349886, A356674.

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

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^k*x)^k/(k!*(1-k^k*x)))))

E.g.f.: Sum_{k>=0} (k^k * x)^k / (k! * (1 - k^k * x)).

A355465 Expansion of Sum_{k>=0} (k^k * x/(1 - k^k * x))^k.

Original entry on oeis.org

1, 1, 17, 19812, 4296562388, 298027622009561768, 10314429455106223377205859112, 256923580408437742134605162130019436138968, 6277101736867794060924264576844540796924098543875220742528

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A349886, A355466.

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, (k^k*x/(1-k^k*x))^k))

a(n) = if(n==0, 1, sum(k=1, n, k^(k*n)*binomial(n-1, k-1)));

a(n) = Sum_{k=1..n} k^(k*n) * binomial(n-1,k-1) for n > 0.

cp's OEIS Frontend

A349893 a(n) = Sum_{k=0..n} k^(k*(n-k)).

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A349891 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(k*n).

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Crossrefs

Programs

PARI

PARI

Formula

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

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PARI

PARI

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A355466 Expansion of Sum_{k>=0} (k^k * x)^k/(1 - k^k * x)^(k+1).

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PARI

PARI

PARI

Formula

A356689 a(n) = n! * Sum_{k=0..n} k^(k*n)/k!.

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PARI

PARI

Formula

A355465 Expansion of Sum_{k>=0} (k^k * x/(1 - k^k * x))^k.

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Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula