A349878 -id:A349878 - 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

A349879 Expansion of Sum_{k>=0} k^4 * x^k/(1 - k * x).

Original entry on oeis.org

0, 1, 17, 114, 564, 2507, 10961, 49260, 231928, 1150781, 6017297, 33085294, 190777804, 1150650935, 7241707281, 47454741400, 323154690928, 2282779984281, 16700904481425, 126356632381834, 987303454919204, 7957133905597635, 66071772829234641

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

In general, for s>=1, Sum_{k=0..n} k^(n-k+s) ~ sqrt(2*Pi) * ((n + s)/LambertW(exp(1)*(n + s)))^(1/2 + (n + s)*(1 - 1/LambertW(exp(1)*(n + s)))) / sqrt(1 + LambertW(exp(1)*(n + s))). - Vaclav Kotesovec, Dec 04 2021

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

Cf. A026898, A003101, A062809, A349878.

Cf. A349855.

Table[Sum[k^(n - k + 4), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, s=4, 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^4*x^k/(1-k*x))))

a(n) = Sum_{k=0..n} k^(n-k+4).

a(n) ~ sqrt(2*Pi) * ((n + 4)/LambertW(exp(1)*(n + 4)))^(1/2 + (n + 4)*(1 - 1/LambertW(exp(1)*(n + 4)))) / sqrt(1 + LambertW(exp(1)*(n + 4))). - Vaclav Kotesovec, Dec 04 2021

cp's OEIS Frontend

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

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A349879 Expansion of Sum_{k>=0} k^4 * x^k/(1 - k * x).

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