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A187009 G.f. A(x) satisfies: [x^(n+1)] A(F^n(x)) = 0 for n>0 where F^n(x) denotes the n-th iteration of F(x) = x+x^2 with F^0(x)=x.

%I A187009 #6 Nov 21 2024 16:13:17
%S A187009 1,-1,2,-6,20,-80,348,-1778,9892,-64392,449596,-3609782,30152616,
%T A187009 -284037468,2694480888,-28592860322,295151311376,-3440953545088,
%U A187009 37165311149276,-471576198145144,5062381083026352,-71104461751595892
%N A187009 G.f. A(x) satisfies: [x^(n+1)] A(F^n(x)) = 0 for n>0 where F^n(x) denotes the n-th iteration of F(x) = x+x^2 with F^0(x)=x.
%e A187009 G.f.: A(x) = x - x^2 + 2*x^3 - 6*x^4 + 20*x^5 - 80*x^6 + 348*x^7 +...
%e A187009 Let F^n(x) denote the n-th iteration of F(x) = x+x^2 with F^0(x)=x,
%e A187009 then the table of coefficients in A(F^n(x)), n>=0, begins:
%e A187009 [1, -1, 2, -6, 20, -80, 348, -1778, 9892, -64392, ...];
%e A187009 [1, 0, 0, -1, 2, -14, 44, -348, 1476, -14148, 73920, ...];
%e A187009 [1, 1, 0, -1, -2, -10, -24, -231, -654, -9276, -32456, ...];
%e A187009 [1, 2, 2, 0, -6, -26, -108, -570, -3216, -22622, -162596, ...];
%e A187009 [1, 3, 6, 8, 0, -54, -324, -1776, -10594, -71702, -540448, ...];
%e A187009 [1, 4, 12, 29, 50, 0, -616, -4846, -32686, -228926, -1749972, ...];
%e A187009 [1, 5, 20, 69, 202, 436, 0, -8629, -84140, -680298, -5508864, ...];
%e A187009 [1, 6, 30, 134, 538, 1880, 4912, 0, -143442, -1672428, -15821492, ...];
%e A187009 [1, 7, 42, 230, 1164, 5404, 22108, 68098, 0, -2762748, -37526484, ...];
%e A187009 [1, 8, 56, 363, 2210, 12646, 67092, 315784, 1122952, 0, -60534272, ..];
%e A187009 [1, 9, 72, 539, 3830, 25930, 166520, 997581, 5322126, 21488640, 0, ..]; ...
%e A187009 in which the main diagonal equals all zeros after the initial '1';
%e A187009 the lower triangular portion of the above table forms triangle A187005.
%o A187009 (PARI) {ITERATE(F,n,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
%o A187009 {a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=-Vec(subst(x*Ser(A),x,ITERATE(x+x^2,i,#A)))[#A]);A[n]}
%Y A187009 Cf. A187005, A135080, A122888, A135081, A187006, A187007, A187008.
%K A187009 sign
%O A187009 1,3
%A A187009 _Paul D. Hanna_, Mar 02 2011