A193563 -id:A193563 - cp's OEIS frontend

A368772 a(n) = (n!)^3 * Sum_{k=0..n} (k/k!)^3.

0, 1, 16, 459, 29440, 3680125, 794907216, 272653175431, 139598425821184, 101767252423643865, 101767252423643866000, 135452212975869985647331, 234061424022303335198589696, 514232948577000427431301564309, 1411055210895289172871491492466640

Offset: 0

Seiichi Manyama, Jan 05 2024

Cf. A368769, A368770, A368771.

Cf. A193563, A217284.

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

a(0) = 0; a(n) = n^3 * a(n-1) + n^3.
a(n) = n^3 * A217284(n-1) for n > 0.
a(n) = Sum_{k=1..n} (k!*binomial(n,k))^3. - Ridouane Oudra, Jun 14 2025

A346410 a(n) = (n!)^2 * Sum_{k=0..n-1} 1 / ((n-k) * k!)^2.

Original entry on oeis.org

0, 1, 5, 22, 152, 2001, 45097, 1527506, 71864928, 4466430513, 353828600029, 34770661312190, 4148422395161464, 590479899466175681, 98824492409739430401, 19209838771051338898234, 4291488438323868507946880, 1091819942877526843993466529, 313819508664449992611846900981

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

nonn

Cf. A002104, A006040, A066998, A193563, A336291, A346411.

Table[(n!)^2 Sum[1/((n - k) k!)^2, {k, 0, n - 1}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[PolyLog[2, x] BesselI[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = polylog(2,x) * BesselI(0,2*sqrt(x)).

cp's OEIS Frontend

A368772 a(n) = (n!)^3 * Sum_{k=0..n} (k/k!)^3.

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