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EXAMPLE
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E.g.f.: A(x) = x + 2*x^2/2! - 12*x^3/3! + 180*x^4/4! - 4080*x^5/5! + 113400*x^6/6! - 3281040*x^7/7! + 75675600*x^8/8! - 207930240*x^9/9! - 54890589600*x^10/10! +...
The coefficients of x^k/k!, k>=1, in the n-th iteration of e.g.f. A(x) begin:
n=1: [1, 2, -12, 180, -4080, 113400, -3281040, 75675600, ...];
n=2: [1, 4, -12, 144, -2520, 41760, 186480, -86446080, ...];
n=3: [1, 6, 0, 36, -360, -16200, 1663200, -91813680, ...];
n=4: [1, 8, 24, 0, 240, -21600, 1058400, -25482240, ...];
n=5: [1, 10, 60, 180, 0, -9000, 216720, 12615120, ...];
n=6: [1, 12, 108, 720, 2520, 0, -166320, 10967040, ...];
n=7: [1, 14, 168, 1764, 14280, 83160, 0, -8431920, ...];
n=8: [1, 16, 240, 3456, 44640, 504000, 4374720, 0, ...];
n=9: [1, 18, 324, 5940, 105840, 1798200, 27987120, 343722960, 0, ...]; ...
where the coefficient of x^n in the n-th iteration = 0 for n>2.
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PROG
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(PARI) {ITERATE(n, F)=local(G=x); for(i=1, n, G=subst(G, x, F)); G}
{a(n)=local(A=[1, 2]); for(m=3, n, A=concat(A, 0); A[#A]=-(#A-1)!*Vec(ITERATE(#A, sum(k=1, #A-1, A[k]*x^k/k!)+x*O(x^#A)))[#A]); A[n]}
for(n=1, 30, print1(a(n), ", "))
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