A355575 -id:A355575 - cp's OEIS frontend

A358081 Expansion of e.g.f. 1/(1 - x^3 * exp(x)).

1, 0, 0, 6, 24, 60, 840, 10290, 80976, 847224, 13306320, 190271070, 2677088040, 46082426676, 874515884424, 16582066303530, 336875275380000, 7539189088358640, 176554878235711776, 4295134487197296054, 111114287924643309240, 3036073975138066955820

Offset: 0

Seiichi Manyama, Oct 30 2022

Seiichi Manyama, Table of n, a(n) for n = 0..431

Cf. A006153, A358080.

Cf. A292889, A355575, A358065.

With[{nn=30},CoefficientList[Series[1/(1-x^3 Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*exp(x))))

a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(n-3*k)!);

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/(n - 3*k)!.

a(n) ~ n! / ((1 + LambertW(1/3)) * 3^(n+1) * LambertW(1/3)^n). - Vaclav Kotesovec, Oct 30 2022

A345747 a(n) = n! * Sum_{k=0..floor(n/2)} k^(n - 2*k)/k!.

Original entry on oeis.org

1, 0, 2, 6, 36, 240, 2280, 27720, 425040, 7862400, 171188640, 4319330400, 125199708480, 4142318019840, 155388782989440, 6557345831836800, 308677784640825600, 16079233115648102400, 920518264903690252800, 57603377545940850624000

Offset: 0

Seiichi Manyama, Sep 17 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..315

Cf. A354436, A355575.

Cf. A104872, A216507, A356628.

Join[{1}, Table[n!*Sum[k^(n - 2*k)/k!, {k, 0, n/2}], {n, 1, 20}]] (* Vaclav Kotesovec, Oct 30 2022 *)

PARI
```
a(n) = n!*sum(k=0, n\2, k^(n-2*k)/k!);
```

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

E.g.f.: Sum_{k>=0} x^(2*k) / (k! * (1 - k * x)).

a(n) ~ sqrt(2*Pi) * exp((n - 1/2)/LambertW(exp(2/3)*(2*n - 1)/6) - 2*n) * n^(2*n + 1/2) / (3^(n + 1/2) * sqrt(1 + LambertW(exp(2/3)*(2*n - 1)/6)) * LambertW(exp(2/3)*(2*n - 1)/6)^n). - Vaclav Kotesovec, Oct 30 2022

A352945 a(n) = Sum_{k=0..floor(n/3)} k^(n-3*k).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 3, 5, 10, 20, 42, 93, 214, 516, 1307, 3473, 9659, 28002, 84257, 262229, 842196, 2787864, 9506796, 33388393, 120727844, 449148808, 1717595949, 6743420017, 27147152525, 111931584098, 472225684599, 2037019695797, 8979468552886, 40432306870108

Offset: 0

Seiichi Manyama, Apr 09 2022

Seiichi Manyama, Table of n, a(n) for n = 0..738

Cf. A104872, A355575.

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

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

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

a(n) ~ sqrt(2*Pi/3) * (n/(3*LambertW(exp(1)*n/3)))^(n + 1/2 - n/LambertW(exp(1)*n/3)) / sqrt(1 + LambertW(exp(1)*n/3)). - Vaclav Kotesovec, Apr 14 2022

cp's OEIS Frontend

A358081 Expansion of e.g.f. 1/(1 - x^3 * exp(x)).

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A345747 a(n) = n! * Sum_{k=0..floor(n/2)} k^(n - 2*k)/k!.

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Mathematica

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A352945 a(n) = Sum_{k=0..floor(n/3)} k^(n-3*k).

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