login
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.
2

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

%S 1,1,-6,80,-1570,39914,-1242192,45954240,-1986728358,99266274404,

%T -5684436494176,370411466138116,-27283461238591012,

%U 2257174498490657648,-208468812531599092816,21371789110179099868480

%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 - 6*x^3 + 80*x^4 - 1570*x^5 + 39914*x^6 +...

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

%e [1, 1,. -6,.. 80,. -1570,.. 39914,. -1242192,.. 45954240,...];

%e [1, 2, -10,. 131,. -2582,.. 66364,. -2092772,.. 78479586,...];

%e [1, 3,(-12), 159,. -3182,.. 83072,. -2659272,. 101127541,...];

%e [1, 4,(-12),(170), -3492,.. 92922,. -3022460,. 116545332,...];

%e [1, 5, -10, (170),(-3610),. 98080,. -3241848,. 126687816,...];

%e [1, 6,. -6,. 165, (-3610),(100114), -3360480,. 132988310,...];

%e [1, 7,.. 0,. 161,. -3542, (100114),(-3409000), 136493385,...];

%e [1, 8,.. 8,. 164,. -3432,.. 98812, (-3409000),(137966664),...];

%e [1, 9,. 18,. 180,. -3282,.. 96702,. -3375648, (137966664),...]; ...

%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,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, A177775.

%K sign

%O 1,3

%A _Paul D. Hanna_, May 13 2010