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EXAMPLE
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E.g.f.: A(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 196*x^4/4! + 4652*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS.
If we form an array of coefficients of x^k/k! in A(x)^n, n>=0, like so:
A^0: [1],0, 0, 0, 0, 0, 0, 0, 0, ...;
A^1: [1, 1], 2, 14, 196, 4652, 166168, 8232296, 535974416, ...;
A^2: [1, 2, 6], 40, 528, 11824, 403840, 19373792, 1232259840, ...;
A^3: [1, 3, 12, 84], 1068, 22716, 741456, 34375200, 2132407248, ...;
A^4: [1, 4, 20, 152, 1912], 39008, 1218496, 54513152, 3292657664, ...;
A^5: [1, 5, 30, 250, 3180, 62980],1889080, 81499400, 4785873360, ...;
A^6: [1, 6, 42, 384, 5016, 97632, 2826288],117620256, 6706638336, ...;
A^7: [1, 7, 56, 560, 7588, 146804, 4127200, 165911312], 9177810320, ...;
A^8: [1, 8, 72, 784, 11088, 215296, 5918656, 230372480, 12358846848], ...; ...
then we can illustrate how the sum of the coefficients of x^k, k=0..n, in A(x)^n (shown above in brackets) equals (n+1)!:
1! = 1;
2! = 1 + 1;
3! = 1 + 2 + 6/2!;
4! = 1 + 3 + 12/2! + 84/3!;
5! = 1 + 4 + 20/2! + 152/3! + 1912/4!;
6! = 1 + 5 + 30/2! + 250/3! + 3180/4! + 62980/5!; ...
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PROG
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(PARI) /* By Definition (slow): */
{a(n)=if(n==0, 1, n!*((n+1)! - sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j/j!)^n + x*O(x^k), k)))/n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Faster, using series reversion: */
{a(n)=local(B=sum(k=0, n+1, (k+1)!*x^k)+x^3*O(x^n), G=1+x*O(x^n));
for(i=1, n, G = 1 + intformal( (B-1)*G/x - B*G^2)); n!*polcoeff(x/serreverse(x*G), n)}
for(n=0, 30, print1(a(n), ", "))
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