%I #2 Mar 30 2012 18:37:22

%S 1,1,-4,39,-580,11480,-285116,8617217,-311138320,13245849264,

%T -657721045720,37721447340698,-2476051190767536,184449202720026868,

%U -15472664808232769104,1451318259607442040637,-151254398423642331357224

%N G.f. A(x) satisfies: [x^n] A_{n}(x) = [x^n] A_{n-1}(x) for n>2 where A_{n+1}(x) = A_{n}(A(x)) denotes iteration with A_0(x)=x.

%e G.f.: A(x) = x + x^2 - 4*x^3 + 39*x^4 - 580*x^5 + 11480*x^6 +...

%e Coefficients in the initial iterations of A(x) begin:

%e [1, 1, -4,. 39,.. -580,. 11480,. -285116,.. 8617217,. -311138320,...];

%e [1, 2,(-6), 59,.. -898,. 18228,. -463816,. 14330618,. -527519702,...];

%e [1, 3,(-6),(66), -1048,. 21932,. -572180,. 18055088,. -676555682,...];

%e [1, 4, -4, (66),(-1100), 23750,. -634548,. 20415192,. -777438522,...];

%e [1, 5,. 0,. 65, (-1100),(24430), -666940,. 21835125,. -843666770,...];

%e [1, 6,. 6,. 69,. -1070, (24430),(-679756), 22603642,. -884811200,...];

%e [1, 7, 14,. 84,. -1008,. 24038, (-679756),(22919008), -907726332,...];

%e [1, 8, 24, 116,.. -888,. 23492,. -671320, (22919008),(-917372412),...];

%e [1, 9, 36, 171,.. -660,. 23100,. -656988,. 22701057, (-917372412),...]; ...

%e where the above coefficients in parenthesis illustrate the property

%e that the coefficients of x^n in A_{n}(x) and in A_{n-1}(x) are equal.

%o (PARI) {a(n)=local(F=x+x^2+sum(m=3,n-1,a(m)*x^m)+x*O(x^n),G=x,H); for(i=1,n-1,G=subst(G,x,F));H=subst(G,x,F); if(n<1,0,if(n<3,1,polcoeff(G-H,n)))}

%Y Cf. A177774, A177776.

%K sign

%O 1,3

%A _Paul D. Hanna_, May 13 2010