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 A193202 #8 Dec 31 2012 18:05:04
%S A193202 1,2,0,12,-160,3240,-86688,2922640,-119971584,5847901920,
%T A193202 -332122243200,21653202377664,-1601381638172160,133036354347921024,
%U A193202 -12314128238585510400,1261212911036957548800,-142082122642808666185728,17514853400850824425213440,-2351847513553411263501035520,342599734607249938595012582400
%N A193202 E.g.f. A(x) satisfies: A(A(x)) = x*A'(A(x)).
%H A193202 Paul D. Hanna, <a href="/A193202/b193202.txt">Table of n, a(n) for n = 1..150</a>
%F A193202 E.g.f. satisfies: A( A(x)/A'(x) ) = x.
%F A193202 E.g.f. satisfies: A(x) = Series_Reversion(-G(-x)) where G(x) = -A(-x)/A'(-x) is the e.g.f. of A179420 and satisfies: G(G(x)) = x*G'(x).
%F A193202 The inverse function of A(x), A(x)/A'(x), equals the g.f. of column 0 of the matrix log of the Riordan array (A(x)/x, A(x)).
%F A193202 Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x, then:
%F A193202 _ A(x)/A'(x) = A_{n-1}(x) * A_n(x) / (x * d/dx A_n(x)) for all n.
%e A193202 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^4/4! - 160*x^5/5! + 3240*x^6/6! - 86688*x^7/7! + 2922640*x^8/8! - 119971584*x^9/9! + 5847901920*x^10/10! +...
%e A193202 RELATED EXPANSIONS.
%e A193202 _ A(A(x)) = x + 4*x^2/2! + 12*x^3/3! + 48*x^4/4! + 40*x^5/5! + 2640*x^6/6! - 57456*x^7/7! + 2059904*x^8/8! - 85967136*x^9/9! + 4262310720*x^10/10! +...
%e A193202 _ A'(A(x)) = 1 + 2*x + 4*x^2/2! + 12*x^3/3! + 8*x^4/4! + 440*x^5/5! - 8208*x^6/6! +...
%e A193202 _ A(x)/A'(x) = x - 2*x^2/2! + 12*x^3/3! - 132*x^4/4! + 2200*x^5/5! - 50280*x^6/6! + 1482768*x^7/7! - 54171376*x^8/8! + 2381590944*x^9/9! - 123292821600*x^10/10! +...
%e A193202 Higher order iterations begin:
%e A193202 _ A_3(x) = x + 6*x^2/2! + 36*x^3/3! + 252*x^4/4! + 1800*x^5/5! + 16920*x^6/6! +...
%e A193202 _ A_4(x) = x + 8*x^2/2! + 72*x^3/3! + 768*x^4/4! + 9200*x^5/5! + 126720*x^6/6! +...
%e A193202 _ A_5(x) = x + 10*x^2/2! + 120*x^3/3! + 1740*x^4/4! + 29200*x^5/5! + 561000*x^6/6! +...
%e A193202 Illustrate a main property of the iterations A_n(x) by:
%e A193202 _ A(x)/A'(x) = A(x) * A(A(x)) / (x*d/dx A(A(x)));
%e A193202 _ A(x)/A'(x) = A_2(x) * A_3(x) / (x*d/dx A_3(x));
%e A193202 _ A(x)/A'(x) = A_3(x) * A_4(x) / (x*d/dx A_4(x));
%e A193202 _ A(x)/A'(x) = A_4(x) * A_5(x) / (x*d/dx A_5(x)); ...
%e A193202 which can be shown consistent by the chain rule of differentiation.
%o A193202 (PARI) {a(n)=local(A=x+x^2+sum(m=3, n-1, a(m)*x^m/m!)+x*O(x^n)); if(n<3, n!*polcoeff(A, n), n!*polcoeff(subst(A,x,A)-x*subst(A',x,A), n)/(n-2))}
%Y A193202 Cf. A179420.
%K A193202 sign
%O A193202 1,2
%A A193202 _Paul D. Hanna_, Jul 22 2011