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

%S 1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,50333780640,

%T 1439208976920,45044270036220,1531759925038616,56239576979827360,

%U 2217379518189430404,93441321290076019236,4191262657895865499821

%N a(n) = coefficient of x^n in the (n+1)-th iteration of (x + x^2 + x^3) for n>=1.

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

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

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

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

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

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

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

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

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

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

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

%e F_10:[1, 10, 100, 990, 9720, 94680, 915390, 8787735, (83795085), ...]; ...

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

%e form the initial terms of this sequence.

%o (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)))}

%Y Cf. A166880, A166881, A166882, A166884.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 22 2009