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A187124 G.f. A(x) satisfies: [x^(n+3)] 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 A187124 #10 Jan 27 2025 15:34:30
%S A187124 1,-3,6,-18,48,-195,549,-3465,7452,-112707,-5994,-6866904,-25659292,
%T A187124 -700243362,-5594278734,-106900155574,-1284177510456,-22692117042216,
%U A187124 -348993455353854,-6343625959542180,-114598750263323292,-2239367384128230334,-45442026505346961480,-967951044447512385336
%N A187124 G.f. A(x) satisfies: [x^(n+3)] 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 A187124 G.f.: A(x) = x^3 - 3*x^4 + 6*x^5 - 18*x^6 + 48*x^7 - 195*x^8 +...
%e A187124 Let F^n(x) denote the n-th iteration of F(x) = x+x^2 with F^0(x)=x,
%e A187124 then the table of coefficients in A(F^n(x)), n>=0, begins:
%e A187124   [1, -3, 6, -18, 48, -195, 549, -3465, 7452, -112707, ...];
%e A187124   [1, 0, -3, -5, -12, -72, -333, -2568, -16782, -153204, ...];
%e A187124   [1, 3, 0, -19, -72, -261, -1276, -8079, -58932, -486635, ...];
%e A187124   [1, 6, 15, 0, -174, -1047, -5256, -29676, -202908, -1625427, ...];
%e A187124   [1, 9, 42, 112, 0, -2109, -17211, -112371, -753606, -5711283, ...];
%e A187124   [1, 12, 81, 377, 1128, 0, -31633, -324600, -2614344, -20650886, ...];
%e A187124   [1, 15, 132, 855, 4248, 14373, 0, -564081, -6957390, -66648777, ...];
%e A187124   [1, 18, 195, 1606, 10758, 58269, 221952, 0, -11639502, -167467539,..];
%e A187124   [1, 21, 270, 2690, 22416, 159633, 947117, 4029915, 0, -272551739,...];
%e A187124   [1, 24, 357, 4167, 41340, 359616, 2750067, 17848872, 84135510, 0,...]; ...
%e A187124 in which the main diagonal equals all zeros after the initial '1';
%e A187124 the lower triangular portion of the above table forms triangle A187124.
%o A187124 (PARI) {ITERATE(F, n, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
%o A187124 {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(subst(x^3*Ser(A), x, ITERATE(x+x^2, i, #A)))[#A]); A[n]}
%Y A187124 Cf. A187009, A187119, A187120, A135083, A187121, A187122, A187123.
%K A187124 sign
%O A187124 3,2
%A A187124 _Paul D. Hanna_, Mar 08 2011
%E A187124 More terms from _Michel Marcus_, Jan 27 2025