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 A220379 #7 Dec 13 2012 12:26:12
%S A220379 1,1,3,16,108,836,7136,65708,643522,6638880,71649726,804833052,
%T A220379 9373104396,112833093984,1400634016520,17894022203641,234907670711601,
%U A220379 3164596264280695,43700481254733535,617995260359761384,8942391804824517624,132304132524112742604
%N A220379 G.f. satisfies: A(x) = x + (1-x)*A(A(x))^2.
%F A220379 G.f. A(x) satisfies:
%F A220379 (1) A(x) = x + Sum_{n>=1} (1-x)^n * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
%F A220379 (2) A(x) = x*exp( Sum_{n>=1} (1-x)^n * d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ).
%F A220379 (3) A( (x - A(x)^2) / (1 - A(x)^2) ) = x.
%F A220379 (4) A(A( 1 - 1/((1+x)*(1-A(x)^2)) )) = x.
%F A220379 (5) A(A(x)) = sqrt( (A(x) - x) / (1 - x) ).
%e A220379 G.f.: A(x) = x + x^2 + 3*x^3 + 16*x^4 + 108*x^5 + 836*x^6 + 7136*x^7 +...
%e A220379 The g.f. satisfies the series:
%e A220379 A(x) = x + (1-x)*A(x)^2 + (1-x)^2*d/dx A(x)^4/2! + (1-x)^3*d^2/dx^2 A(x)^6/3! + (1-x)^4*d^3/dx^3 A(x)^8/4! +...
%e A220379 as well as the logarithmic series:
%e A220379 log(A(x)/x) = (1-x)*A(x)^2/x + (1-x)^2*[d/dx A(x)^4/x]/2! + (1-x)^3*[d^2/dx^2 A(x)^6/x]/3! + (1-x)^4*[d^3/dx^3 A(x)^8/x]/4! +...
%e A220379 Related expansions:
%e A220379 A(A(x)) = x + 2*x^2 + 8*x^3 + 48*x^4 + 354*x^5 + 2958*x^6 + 27004*x^7 +...
%e A220379 A(A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 128*x^5 + 964*x^6 + 8100*x^7 +...
%e A220379 (A(x)-x)/(1-x) = x^2 + 4*x^3 + 20*x^4 + 128*x^5 + 964*x^6 + 8100*x^7 +...
%e A220379 The series reversion of the g.f. A(x) equals:
%e A220379 (x-A(x)^2)/(1-A(x)^2) = x - x^2 - x^3 - 6*x^4 - 34*x^5 - 234*x^6 - 1818*x^7 -...
%e A220379 The series reversion of A(A(x)) equals:
%e A220379 1 - 1/((1+x)*(1-A(x)^2)) = x - 2*x^2 - 8*x^4 - 34*x^5 - 242*x^6 - 1852*x^7 -...
%o A220379 (PARI) {a(n)=local(A=x);for(i=1,n,A=x+(1-x)*subst(A,x,A+x*O(x^n))^2);polcoeff(A,n)}
%o A220379 for(n=1,25,print1(a(n),", "))
%o A220379 (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse((x - A^2)/(1-A^2+x*O(x^n)))); polcoeff(A, n))}
%o A220379 for(n=1,25,print1(a(n),", "))
%o A220379 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A220379 {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, (1-x)^m*Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
%o A220379 for(n=1,25,print1(a(n),", "))
%o A220379 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A220379 {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, (1-x)^m*Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
%o A220379 for(n=1,25,print1(a(n),", "))
%Y A220379 Cf. A213591.
%K A220379 nonn
%O A220379 1,3
%A A220379 _Paul D. Hanna_, Dec 13 2012