OFFSET
1,3
COMMENTS
Here the zeroth iteration of A(x) equals x, the first iteration is itself, the 2nd iteration of A(x) = A(A(x)), etc.
EXAMPLE
The coefficients in the n-th iteration of g.f. A(x) begin:
n=1: [1, 1, 2, 12, 138, 2370, 54190, 1553258, 53883088, ...];
n=2: [1, 2, 6, 35, 370, 6000, 132344, 3704032, 126318024, ...];
n=3: [1, 3, 12, 75, 758, 11612, 245746, 6688885, 223699238, ...];
n=4: [1, 4, 20, 138, 1388, 20322, 411708, 10854152, 354952262, ...];
n=5: [1, 5, 30, 230, 2370, 33760, 656414, 16711414, 532707614, ...];
n=6: [1, 6, 42, 357, 3838, 54190, 1018484, 25016120, 775036254, ...];
n=7: [1, 7, 56, 525, 5950, 84630, 1553258, 36874397, 1107956996, ...];
n=8: [1, 8, 72, 740, 8888, 128972, 2337800, 53883088, 1568966580, ...];
n=9: [1, 9, 90, 1008, 12858, 192102, 3476622, 78308058, 2211883428, ...]; ...
where the diagonal of coefficients equals this sequence shift left 1 place.
...
More explicitly, the successive iterations of g.f. A(x) begin:
A(x) = (1)x + x^2 + 2x^3 + 12x^4 + 138x^5 + 2370x^6 + 54190x^7 +...
A(A(x)) = x + (2)x^2 + 6x^3 + 35x^4 + 370x^5 + 6000x^6 + 132344x^7 +...
A(A(A(x))) = x + 3x^2 + (12)x^3 + 75x^4 + 758x^5 + 11612x^6 +...
A(A(A(A(x)))) = x + 4x^2 + 20x^3 + (138)x^4 + 1388x^5 + 20322x^6 +...
A(A(A(A(A(x))))) = x + 5x^2 + 30x^3 + 230x^4 + (2370)x^5 + 33760x^6+...
A(A(A(A(A(A(x)))))) = x + 6x^2 +42x^3 +357x^4 +3838x^5 + (54190)x^6+...
...
PROG
(PARI) {a(n)=local(F=x+x^2+sum(m=3, n-1, a(m)*x^m), G=x+x*O(x^n)); if(n<1, 0, if(n<=2, 1, for(i=1, n-1, G=subst(F, x, G)); return(polcoeff(G, n-1, x))))}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 31 2006
STATUS
approved