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.
lst={};p0=p1=p2=p3=1;Do[p0*=a[n];p1*=p0;p2*=p1;p3*=p2;AppendTo[lst,p3],{n,1,2*3!}];lst
[0] cat [((n^2 -3*n +1)*(&+[Factorial(k): k in [0..(n-1)]]) -(n-2)*Factorial(n) +2*(n-1))/2: n in [1..30]]; // G. C. Greubel, Sep 13 2018
With S[n_]:= Sum[k!, {k,0,n-1}];
Table[Sum[Sum[S[j], {j,0,m-1}], {m,0,n -1}], {n, 0, 30}] (* or *) Table[((n^2 - 3*n + 1)*S[n] - (n - 2)*n! + 2*(n - 1))/2, {n, 0, 30}] (* G. C. Greubel, Sep 13 2018 *)
for(n=0,30, print1(((n^2-3*n+1)*sum(k=0,n-1, k!) - (n-2)*n! + 2*(n -1))/2, ", ")) \\ G. C. Greubel, Sep 13 2018
lst={};p0=1;s0=0;Do[p0*=a[n];s0+=p0;AppendTo[lst,s0],{n,0,4!}];lst
s = 0; lst = {s}; Do[s += BarnesG[n]; AppendTo[lst, s], {n, 2, 13, 1}]; lst (* Zerinvary Lajos, Jul 16 2009 *)
Table[Sum[BarnesG[k+1],{k,1,n}],{n,1,15}] (* Vaclav Kotesovec, Jul 10 2015 *)