This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193563 #35 Jun 19 2025 12:17:02
%S A193563 0,1,8,81,1312,32825,1181736,57905113,3705927296,300180111057,
%T A193563 30018011105800,3632179343801921,523033825507476768,
%U A193563 88392716510763573961,17324972436109660496552,3898118798124673611724425,997918412319916444601453056
%N A193563 a(0) = 0, a(n) = n^2 * (a(n-1) + 1).
%H A193563 Paolo Xausa, <a href="/A193563/b193563.txt">Table of n, a(n) for n = 0..250</a>
%F A193563 From _Seiichi Manyama_, Jan 05 2024: (Start)
%F A193563 a(n) = (n!)^2 * Sum_{k=0..n} (k/k!)^2.
%F A193563 a(n) = n^2 * A006040(n-1) for n > 0. (End)
%F A193563 a(n) = Sum_{k=1..n} (k!*binomial(n,k))^2. - _Ridouane Oudra_, Jun 14 2025
%F A193563 a(n) = n^2 + BesselI(0,2)*(n!)^2 - n^2*hypergeom([1], [n, n], 1) for n > 0. - _Stefano Spezia_, Jun 14 2025
%p A193563 seq(n!^2*add(1/k!^2,k=0..n-1),n=0..16); # _Mark van Hoeij_, May 13 2013
%t A193563 FoldList[#2^2*(# + 1) &, Range[0, 20]] (* _Paolo Xausa_, Jun 18 2025 *)
%o A193563 (PARI) a=[0];for(n=1,20,a=concat(a,(a[#a]+1)*n^2));a \\ _Charles R Greathouse IV_, Jul 31 2011
%Y A193563 Cf. A006040, A007526 (multiply by n instead of n^2), A180255.
%K A193563 nonn,easy
%O A193563 0,3
%A A193563 _Meherzad Lahewala_, Jul 31 2011