A358687 -id:A358687 - cp's OEIS frontend

A356673 a(n) = n! * Sum_{k=0..n} k^(3*(n-k))/k!.

1, 1, 3, 31, 901, 45741, 3960871, 584698843, 130554106761, 40790044059481, 17681098707667531, 10491554658622447191, 8198225417359164798733, 8172446419302496167191941, 10264848632098736708582150511

Offset: 0

Seiichi Manyama, Aug 22 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..185

Cf. A354436, A356672.

Cf. A349880, A356629, A358687.

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

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

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

A358688 a(n) = n! * Sum_{k=0..n} k^(k * (n-k)) / (n-k)!.

Original entry on oeis.org

1, 2, 5, 34, 869, 75866, 28213327, 39049033346, 256215628707257, 7710689746589777938, 1063776147486867074877851, 870059224717752809087935599002, 3104894940194751778363241199111802885, 77521065749331962430758061530260243383954602

Offset: 0

Seiichi Manyama, Nov 26 2022

nonn

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

Cf. A006153, A193421, A358687.

Cf. A349893, A356674.

Table[1 + n!*Sum[k^(k*(n-k))/(n-k)!, {k, 1, n}], {n, 0, 12}] (* Vaclav Kotesovec, Nov 27 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k*exp(x)^k^k)))

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

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

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

cp's OEIS Frontend

A356673 a(n) = n! * Sum_{k=0..n} k^(3*(n-k))/k!.

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A358688 a(n) = n! * Sum_{k=0..n} k^(k * (n-k)) / (n-k)!.

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Mathematica

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