1,3

Let F_n(x) denote the n-th iteration of F(x) = x + x^2 + x^3;

then coefficients in the successive iterations of F(x) begin:

F_0: [(1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];

F(x):[1, (1), 1, 0, 0, 0, 0, 0, 0, 0, 0, ...];

F_2: [1, 2, (4), 6, 8, 8, 6, 3, 1, 0, 0, ...];

F_3: [1, 3, 9, (24), 60, 138, 294, 579, 1053, 1767, 2739, ...];

F_4: [1, 4, 16, 60, (216), 744, 2460, 7818, 23910, 70446, 200160, ...];

F_5: [1, 5, 25, 120, 560, (2540), 11220, 48330, 203230, 835080, ...];

F_6: [1, 6, 36, 210, 1200, 6720, (36930), 199365, 1058175, ...];

F_7: [1, 7, 49, 336, 2268, 15078, 98826, (639093), 4080531, ...];

F_8: [1, 8, 64, 504, 3920, 30128, 228984, 1722084, (12821788),...];

F_9: [1, 9, 81, 720, 6336, 55224, 477000, 4085028, 34700940, (292495896), ...]; ...

where the coefficients along the diagonal (shown above in parenthesis)

form the initial terms of this sequence.

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

Cf. A166880, A166882, A166883, A166884.

Sequence in context: A162314 A112141 A077555 this_sequence A145237 A034256 A074598

Adjacent sequences: A166878 A166879 A166880 this_sequence A166882 A166883 A166884

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 22 2009