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EXAMPLE
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G.f.: 1 + x + x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 245*x^6 + 2072*x^7 +...
such that
A(x) = 1 + x*B(x), B(x) = 1 + x*C(x)^2, C(x) = 1 + x*D(x)^3, D(x) = 1 + x*E(x)^4, E(x) = 1 + x*F(x)^5, F(x) = 1 + x*G(x)^6, G(x) = 1 + x*H(x)^7, ...
where A(x), B(x), C(x), ... are the g.f. of the sequences given below.
A: [1, 1, 1, 2, 7, 36, 245, 2072, 20913, 245012, ...];
B: [1, 1, 2, 7, 36, 245, 2072, 20913, 245012, 3265581, ...];
C: [1, 1, 3, 15, 103, 888, 9147, 109150, 1477575, 22349316, ...];
D: [1, 1, 4, 26, 224, 2351, 28760, 399314, 6183132, 105455687, ...];
E: [1, 1, 5, 40, 415, 5145, 73121, 1162620, 20358145, 388334030, ...];
F: [1, 1, 6, 57, 692, 9906, 160656, 2884554, 56502264, 1195386975, ...];
G: [1, 1, 7, 77, 1071, 17395, 317303, 6357267, 137950303, 3211604480, ...];
H: [1, 1, 8, 100, 1568, 28498, 577808, 12788776, 304827080, 7753676623, ...];
I: [1, 1, 9, 126, 2199, 44226, 987021, 23928972, 621887265, 17173176273, ...]; ...
FIRST DERIVATIVES OF SERIES:
A' = B + x*C^2 + 2!*x^2*C*D^3 + 3!*x^3*C*D^2*E^4 + 4!*x^4*C*D^2*E^3*F^5 + 5!*x^5*C*D^2*E^3*F^4*G^6 + 6!*x^6*C*D^2*E^3*F^4*G^5*H^7 +...
B' = C^2 + 2!*x*C*D^3 + 3!*x^2*C*D^2*E^4 + 4!*x^3*C*D^2*E^3*F^5 + 5!*x^4*C*D^2*E^3*F^4*G^6 + 6!*x^5*C*D^2*E^3*F^4*G^5*H^7 +...
2!*C' = 2!*D^3 + 3!*x*D^2*E^4 + 4!*x^2*D^2*E^3*F^5 + 5!*x^3*D^2*E^3*F^4*G^6 + 6!*x^4*D^2*E^3*F^4*G^5*H^7 + 7!*x^5*D^2*E^3*F^4*G^5*H^6*I^8 +...
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PROG
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(PARI) {a(n)=local(A); A=1+x+x*O(x^n); for(j=0, n-1, A=1+x*A^(n-j)); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Print Row r in Table (this Sequence is at r=1) */
{a(n, r=1)=local(A=vector(3*n+2*r+2, i, 1+x));
for(m=1, 2*n+r, for(j=0, n+r+m, A[n+r+m-j+1]=1+x*(A[n+r+m-j+2] +x^r*O(x^n))^(n+r+m-j+1) ); ); polcoeff(A[r], n)}
for(n=0, 20, print1(a(n, 1), ", "))
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