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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 127*x^4 + 1321*x^5 + 16680*x^6 +...
The coefficients in A(x)^n begin:
n=1: [1, 1, 3, 16, 127, 1321, 16680, 244518, 4049199, ...];
n=2: [1, 2, 7, 38, 295, 2992, 37020, 534386, 8745915, ...];
n=3: [1, 3, 12, 67, 513, 5088, 61716, 877053, 14181891, ...];
n=4: [1, 4, 18, 104, 791, 7696, 91582, 1281160, 20462071, ...];
n=5: [1, 5, 25, 150, 1140, 10916, 127565, 1756710, 27706465, ...];
n=6: [1, 6, 33, 206, 1572, 14862, 170761, 2315256, 36052245, ...];
n=7: [1, 7, 42, 273, 2100, 19663, 222432, 2970108, 45656093, ...];
n=8: [1, 8, 52, 352, 2738, 25464, 284024, 3736560, 56696823, ...];
n=9: [1, 9, 63, 444, 3501, 32427, 357186, 4632138, 69378300, ...]; ...
where the initial terms are derived from the above coefficients like so:
a(1) = 1 = 1;
a(2) = 1 + 2 = 3;
a(3) = 1 + 3 + 12 = 16;
a(4) = 1 + 4 + 18 + 104 = 127;
a(5) = 1 + 5 + 25 + 150 + 1140 = 1321;
a(6) = 1 + 6 + 33 + 206 + 1572 + 14862 = 16680; ...
RELATED EXPANSIONS.
The series B(x) = Series_Reversion(x/A(x)) begins:
B(x) = x + x^2 + 4*x^3 + 26*x^4 + 228*x^5 + 2477*x^6 + 31776*x^7 +...
such that A(x) = 1 + x*B'(x)/(1 - B(x)); also,
B(x) = Sum_{n>=1} b(n)*x^n where b(n) = [x^(n-1)] A(x)^n/n for n>=1:
[1/1, 2/2, 12/3, 104/4, 1140/5, 14862/6, 222432/7, 3736560/8, ...].
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PROG
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(PARI) {a(n)=local(A); if(n<2, n>=0, A=1+x; for(i=2, n, A+=x^i*subst(Pol((A+O(x^i))^i), x, 1)); polcoeff(A, n))}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* A(x) = 1 + x*B'(x)/(1 - B(x)) where B(x/A(x)) = x: */
{a(n)=local(A=1+x); for(i=1, n, B=serreverse(x/A+x*O(x^n)); A=1+x*deriv(B)/(1-B)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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