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.
E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +...
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 78*x^3/3! + 1648*x^4/4! + 49500*x^5/5! +... Related expansions: . x*A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! +... . x*A(x)^2 = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +.. . x*A'(x) = x + 12*x^2/2! + 234*x^3/3! + 6592*x^4/4! + 247500*x^5/5! +... . A(x*A(x)) = 1 + x + 8*x^2/2! + 132*x^3/3! + 3400*x^4/4! + 120940*x^5/5! +... . A(x*A(x))^2 = 1 + 2*x + 18*x^2/2! + 312*x^3/3! + 8240*x^4/4! + 297000*x^5/5! +... Illustrate the iterations G_n(x) of G(x) = x*A(x) by: . [G_3(x)/x]^2 = A(x)^2 * G_2'(x); . [G_4(x)/x]^2 = A(x)^2 * G_3'(x); . [G_5(x)/x]^2 = A(x)^2 * G_4'(x); ... which can be shown by the chain rule of differentiation. ... The RIORDAN ARRAY (A(x), x*A(x)) begins: . 1; . 1, 1; . 6/2!, 2, 1; . 78/3!, 14/2!, 3, 1; . 1648/4!, 192/3!, 24/2!, 4, 1; . 49500/5!, 4136/4!, 348/3!, 36/2!, 5, 1; . 1957968/6!, 124840/5!, 7680/4!, 552/3!, 50/2!, 6, 1; . 97097336/7!, 4928256/6!, 233940/5!, 12520/4!, 810/3!, 66/2!, 7, 1; ... where the g.f. of column k = A(x)^(k+1) for k>=0. ... The MATRIX LOG of the above Riordan array (A(x), x*A(x)) begins: . 0; . 1, 0; . 4/2!, 2, 0; . 42/3!, 8/2!, 3, 0; . 768/4!, 84/3!, 12/2!, 4, 0; . 20680/5!, 1536/4!, 126/3!, 16/2!, 5, 0; . 749040/6!, 41360/5!, 2304/4!, 168/3!, 20/2!, 6, 0; . 34497792/7!, 1498080/6!, 62040/5!, 3072/4!, 210/3!, 24/2!, 7, 0; ... where the g.f. of column k = (k+1)*x*A(x)^2 for k>=0.
{a(n)=local(A=1+x+sum(m=2,n-1,a(m)*x^m/m!)+x*O(x^(n+5)));if(n<2,n!*polcoeff(A,n),n!*polcoeff(subst(A,x,x*A)^2,n)/(n-1))}
E.g.f: A(x) = x + 6*x^3/3! + 540*x^5/5! + 156240*x^7/7! + 96480720*x^9/9! + 104661849600*x^11/11! + 177947471782080*x^13/13! + 439942718370355200*x^15/15! +... Related expansions: A(x)^3 = 6*x^3/3! + 360*x^5/5! + 83160*x^7/7! + 43908480*x^9/9! +... A(A(x)) = x + 12*x^3/3! + 1440*x^5/5! + 509040*x^7/7! + 368686080*x^9/9! +... A(A(x))^3 = 6*x^3/3! + 720*x^5/5! + 241920*x^7/7! + 165110400*x^9/9! +... A'(x) = 1 + 6*x^2/2! + 540*x^4/4! + 156240*x^6/6! + 96480720*x^8/8! +...
/* Coefficients of A_m(x) = m-th iteration of A(x): */
{a(n,m=1)=local(A=x+x^3,D);for(i=1,n,D=x;A=x+sum(k=1,n,m^k*(D=(A+x*O(x^n))^3*deriv(D))/k!));if(n<1,0,n!*polcoeff(A,n))}