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%I #7 Mar 07 2018 18:21:27
%S 1,2,14,176,3256,80464,2508432,94989088,4253855744,220832422912,
%T 13086217711232,874394085363328,65223621800358400,5386555341558256640,
%U 489099399479737741824,48537534403454393734656,5237456183532299486332928,611754844797595116933365760,77043077295500803834985809920,10424760435433013614451330201600
%N E.g.f. A(x) satisfies: A'(x) = 1/(1 - A(A(x)))^2.
%F E.g.f. A(x) satisfies:
%F (1) A(x) = Series_Reversion( Integral (1 - A(x))^2 dx ).
%F (2) A'(x) = 1 / (1 - A(A(x)))^2.
%F (3) A''(x) = 2 / ( (1 - A(A(x)))^5 * (1 - A(A(A(x))))^2 ).
%e G.f.: A(x) = x + 2*x^2/2! + 14*x^3/3! + 176*x^4/4! + 3256*x^5/5! + 80464*x^6/6! + 2508432*x^7/7! + 94989088*x^8/8! + 4253855744*x^9/9! + ...
%e such that A'(x) = 1/(1 - A(A(x)))^2.
%e RELATED SERIES.
%e (1 - A(x))^2 = 1 - 2*x - 2*x^2/2! - 16*x^3/3! - 216*x^4/4! - 4192*x^5/5! - 107376*x^6/6! - 3444384*x^7/7! - 133557568*x^8/8! - 6103032256*x^9/9! + ...
%e Series_Reversion(A(x)) = x - 2*x^2/2! - 2*x^3/3! - 16*x^4/4! - 216*x^5/5! - 4192*x^6/6! - 107376*x^7/7! + ...
%e A(A(x)) = x + 4*x^2/2! + 40*x^3/3! + 656*x^4/4! + 15152*x^5/5! + 455280*x^6/6! + 16947776*x^7/7! + 756120640*x^8/8! + 39475100864*x^9/9! + ...
%o (PARI) {a(n)=local(A=x, G); for(i=1, n, G = intformal( (1 - A +x*O(x^n))^2 ); A = serreverse(G)); n!*polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) {a(n) = local(A=x); for(i=1, n, A = serreverse(intformal( (1-A +x*O(x^n))^2 ))); n!*polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A210949.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Mar 07 2018
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