A349879 -id:A349879 - cp's OEIS frontend

A062809 a(n) = Sum_{i = 1..n} (n - i)^(1 + i).

0, 1, 5, 18, 60, 203, 725, 2772, 11368, 49853, 232757, 1151902, 6018772, 33087191, 190780197, 1150653904, 7241710912, 47454745785, 323154696165, 2282779990474, 16700904488684, 126356632390275, 987303454928949

Offset: 1

Olivier Gérard, Jun 23 2001

Seiichi Manyama, Table of n, a(n) for n = 1..600

Cf. A003101, A349878, A349879.

a:=n->sum((n-j)^j,j=2..n): seq(a(n), n=2..24); # Zerinvary Lajos, Jun 07 2008

Mathematica
```
Sum[(n - i)^(1 + i), {i, 1, n}]
```

a(n) = sum(i=1, n, (n-i)^(1+i)); \\ Michel Marcus, Mar 24 2019

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

A349878 Expansion of Sum_{k>=0} k^3 * x^k/(1 - k * x).

Original entry on oeis.org

0, 1, 9, 44, 178, 689, 2723, 11304, 49772, 232657, 1151781, 6018628, 33087022, 190780001, 1150653679, 7241710656, 47454745496, 323154695841, 2282779990113, 16700904488284, 126356632389834, 987303454928465, 7957133905608283, 66071772829246808

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A026898, A003101, A062809, A349879.

Cf. A349854.

Table[Sum[k^(n - k + 3), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, s=3, t=1) = sum(k=0, n, k^(t*(n-k)+s));

my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^3*x^k/(1-k*x))))

a(n) = Sum_{k=0..n} k^(n-k+3).

a(n) ~ sqrt(2*Pi) * ((n + 3)/LambertW(exp(1)*(n + 3)))^(1/2 + (n + 3)*(1 - 1/LambertW(exp(1)*(n + 3)))) / sqrt(1 + LambertW(exp(1)*(n + 3))). - Vaclav Kotesovec, Dec 04 2021

cp's OEIS Frontend

A062809 a(n) = Sum_{i = 1..n} (n - i)^(1 + i).

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A349878 Expansion of Sum_{k>=0} k^3 * x^k/(1 - k * x).

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Mathematica

PARI

PARI

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