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 A061573 #24 Dec 23 2024 13:09:31
%S A061573 1,6,60,984,24720,890640,43646400,2793409920,226266566400,
%T A061573 22626660268800,2737825932441600,394246934750592000,
%U A061573 66627731979077068800,13059035467986283776000,2938282980298221523968000,752200442956365632925696000,217385928014390023602954240000
%N A061573 a(n) = (n!)^2*Sum_{k=1..n} 1/k!.
%H A061573 Harry J. Smith, <a href="/A061573/b061573.txt">Table of n, a(n) for n = 1..100</a>
%F A061573 Recurrence: a(1) = 1, a(2) = 6, a(n) = n*(n+1)*a(n-1) - n*(n-1)^2*a(n-2) for n >=3. The sequence b(n) = n!^2 also satisfies this recurrence with the initial conditions b(1) = 1 and b(2) = 4. Hence we have the finite continued fraction expansion a(n)/b(n) = 1/(1-2/(6-12/(12-...-n*(n-1)^2/(n*(n+1))))). Lim_{n -> infinity} a(n)/b(n) = e - 1 = 1/(1-2/(6-12/(12-...-n*(n-1)^2/(n*(n+1))-...))) = 1/(1-1/(3-2/(4-...-n/(n+2)-...))). Cf. A000522 and A061572. - _Peter Bala_, Jul 10 2008
%F A061573 a(n) = n!*A002627(n). - _R. J. Mathar_, Mar 18 2017
%F A061573 Sum_{n>=1} a(n) * x^n / (n!)^2 = (exp(x) - 1) / (1 - x). - _Ilya Gutkovskiy_, Jul 15 2021
%t A061573 Table[(n!)^2 Sum[1/k!,{k,n}],{n,20}] (* _Harvey P. Dale_, Dec 02 2021 *)
%o A061573 (PARI) a(n) = { n!^2*sum(k=1, n, 1/k!) } \\ _Harry J. Smith_, Jul 24 2009
%Y A061573 Cf. A000522, A002627, A061572.
%K A061573 nonn
%O A061573 1,2
%A A061573 _N. J. A. Sloane_, May 19 2001