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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 121*x^4 + 1166*x^5 + 13508*x^6 + ...
Related expansions:
A(x*A(x)) = 1 + x + 4*x^2 + 25*x^3 + 206*x^4 + 2077*x^5 + 24603*x^6 + ...
A'(x*A(x)) = 1 + 6*x + 54*x^2 + 598*x^3 + 7714*x^4 + 112726*x^5 + ...
A(x)*A(A(x)) = 1 + 2*x + 8*x^2 + 48*x^3 + 380*x^4 + 3709*x^5 + 42856*x^6 + ...
A(x)^2*A'(A(x)) = 1 + 8*x + 73*x^2 + 786*x^3 + 9799*x^4 + 138760*x^5 + ...
where A(x) = 1 + x*A(x)*A(x*A(x)) + x^2*A(x)^2*A'(x*A(x)).
Let G(x) = 1 + x*A(x) + x^2*A'(x) be the g.f. of A113882;
the table of coefficients in G(x)^n begins:
G(x)^1: [(1), 1, 2, 9, 64, 605, 6996, 94556, ...];
G(x)^2: [1, (2), 5, 22, 150, 1374, 15539, 206676, ...];
G(x)^3: [1, 3, (9), 40, 264, 2346, 25937, 339294, ...];
G(x)^4: [1, 4, 14, (64), 413, 3568, 38558, 495848, ...];
G(x)^5: [1, 5, 20, 95, (605), 5096, 53840, 680365, ...];
G(x)^6: [1, 6, 27, 134, 849, (6996), 72302, 897558, ...];
G(x)^7: [1, 7, 35, 182, 1155, 9345, (94556), 1152936, ...]; ...
where the terms in parentheses form the initial terms of this sequence:
[(1)/1, (2)/2, (9)/3, (64)/4, (605)/5, (6996)/6, (94556)/7, ...].
Further, the logarithm of the g.f. A(x) may be formed from a diagonal in the above table:
log(A(x)) = x + 5*x^2/2 + 40*x^3/3 + 413*x^4/4 + 5096*x^5/5 + 72302*x^6/6 + 1152936*x^7/7 + ...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A*subst(A, x, x*A+x*O(x^n))+x^2*A^2*subst(A', x, x*A+x*O(x^n))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=local(G=1+x); for(k=0, n, G=1+x*deriv(serreverse(x/(G+x^2*O(x^n))))); polcoeff(serreverse(x/G)/x, n)}
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