A180255 -id:A180255 - cp's OEIS frontend

A368770 a(n) = (n!)^3 * Sum_{k=0..n} k/(k!)^3.

0, 1, 10, 273, 17476, 2184505, 471853086, 161845608505, 82864951554568, 60408549683280081, 60408549683280081010, 80403779628445787824321, 138937731197954321360426700, 305246195441905644028857459913, 837595560292589087215184870001286

Offset: 0

Seiichi Manyama, Jan 04 2024

nonn

Cf. A368769, A368771, A368772.

Cf. A180255, A368776.

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

a(0) = 0; a(n) = n^3 * a(n-1) + n.

a(n) = n * A368776(n-1) for n > 0.

A193563 a(0) = 0, a(n) = n^2 * (a(n-1) + 1).

Original entry on oeis.org

0, 1, 8, 81, 1312, 32825, 1181736, 57905113, 3705927296, 300180111057, 30018011105800, 3632179343801921, 523033825507476768, 88392716510763573961, 17324972436109660496552, 3898118798124673611724425, 997918412319916444601453056

Offset: 0

Meherzad Lahewala, Jul 31 2011

Paolo Xausa, Table of n, a(n) for n = 0..250

Cf. A006040, A007526 (multiply by n instead of n^2), A180255.

seq(n!^2*add(1/k!^2,k=0..n-1),n=0..16);   # Mark van Hoeij, May 13 2013

FoldList[#2^2*(# + 1) &, Range[0, 20]] (* Paolo Xausa, Jun 18 2025 *)

a=[0];for(n=1,20,a=concat(a,(a[#a]+1)*n^2));a \\ Charles R Greathouse IV, Jul 31 2011

From Seiichi Manyama, Jan 05 2024: (Start)
a(n) = (n!)^2 * Sum_{k=0..n} (k/k!)^2.
a(n) = n^2 * A006040(n-1) for n > 0. (End)
a(n) = Sum_{k=1..n} (k!*binomial(n,k))^2. - Ridouane Oudra, Jun 14 2025
a(n) = n^2 + BesselI(0,2)*(n!)^2 - n^2*hypergeom([1], [n, n], 1) for n > 0. - Stefano Spezia, Jun 14 2025

cp's OEIS Frontend

A368770 a(n) = (n!)^3 * Sum_{k=0..n} k/(k!)^3.

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