A193563 a(0) = 0, a(n) = n^2 * (a(n-1) + 1).
0, 1, 8, 81, 1312, 32825, 1181736, 57905113, 3705927296, 300180111057, 30018011105800, 3632179343801921, 523033825507476768, 88392716510763573961, 17324972436109660496552, 3898118798124673611724425, 997918412319916444601453056
Offset: 0
Links
- Paolo Xausa, Table of n, a(n) for n = 0..250
Programs
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Maple
seq(n!^2*add(1/k!^2,k=0..n-1),n=0..16); # Mark van Hoeij, May 13 2013
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Mathematica
FoldList[#2^2*(# + 1) &, Range[0, 20]] (* Paolo Xausa, Jun 18 2025 *)
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PARI
a=[0];for(n=1,20,a=concat(a,(a[#a]+1)*n^2));a \\ Charles R Greathouse IV, Jul 31 2011
Formula
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