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.
At n=7, the 7th term of A(x)^7 is 7! x^6, as demonstrated by A(x)^7 = 1 + 7 x + 28 x^2 + 91 x^3 + 294 x^4 + 1092 x^5 + 5040 x^6 + 29093 x^7 + 203651 x^8 + ... . A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 206*x^6 + ... = x/series_reversion(x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 120*x^6 + ...). Related expansions: log(A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 + ...; 1 - x/(A(x) - 1) = x + x^2 + 4*x^3 + 21*x^4 + 136*x^5 + 1030*x^6 +...; (d/dx)((A(x) - 1)/x) = 1 + 4*x + 21*x^2 + 136*x^3 + 1030*x^4 + ... .
a = ConstantArray[0,20]; a[[1]]=1; a[[2]]=1; a[[3]]=2; Do[a[[n]] = (n-1)*a[[n-1]] + Sum[(j-1)*a[[j]]*a[[n-j]],{j,2,n-2}],{n,4,20}]; Flatten[{1,a}] (* Vaclav Kotesovec after David Callan, Feb 22 2014 *)
InverseSeries[Series[Exp[-x] ExpIntegralEi[x], {x, Infinity, 20}]][[3]] (* Vladimir Reshetnikov, Apr 24 2016 *)
a(n)=if(n<0,0,if(n<=1,1,(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j));))
a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,(k-1)!))))[n+1] \\ Paul D. Hanna, Jul 09 2006
{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/(1-x*deriv(A)/A));polcoeff(A,n)}
{a(n)=local(F=1+x*O(x^n)); for(i=0,n,F=1+x*F+x^2*F*deriv(F)+x*O(x^n));polcoeff(1+x*F,n)} \\ Paul D. Hanna, Sep 02 2008
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