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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 118*x^5 + 674*x^6 +...
Illustration of logarithmic derivation.
If we form an array of coefficients of x^k in A(x)^n, n>=1, like so:
A^1: [1],1, 2, 6, 24, 118, 674, 4308, ...;
A^2: [1, 2], 5, 16, 64, 308, 1716, 10724, ...;
A^3: [1, 3, 9], 31, 126, 600, 3278, 20070, ...;
A^4: [1, 4, 14, 52], 217, 1032, 5560, 33440, ...;
A^5: [1, 5, 20, 80, 345], 1651, 8820, 52270, ...;
A^6: [1, 6, 27, 116, 519, 2514],13385, 78420, ...;
A^7: [1, 7, 35, 161, 749, 3689, 19663], 114269, ...; ...
then the sums of the coefficients of x^k, k=0..n-1, in A(x)^n (shown above in brackets) begin:
1 = 1;
1 + 2 = 3;
1 + 3 + 9 = 13;
1 + 4 + 14 + 52 = 71;
1 + 5 + 20 + 80 + 345 = 451;
1 + 6 + 27 + 116 + 519 + 2514 = 3183;
1 + 7 + 35 + 161 + 749 + 3689 + 19663 = 24305; ...
and equal the coefficients in log(A(x)):
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 71*x^4/4 + 451*x^5/5 + 3183*x^6/6 + 24305*x^7/7 + 197551*x^8/8 +...
The main diagonal in the above table forms the g.f. G(x) of A088714:
[1/1, 2/2, 9/3, 52/4, 345/5, 2514/6, 19663/7, ...]
where G(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 419*x^5 + 2809*x^6 +...
satisfies A'(x)/A(x) = (G(x) + x*G'(x)) / (1 - x*G(x)).
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PROG
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(PARI) a(n)=local(A=1+x); for(i=1, n, A=(1+A*serreverse(x/(A+x*O(x^n))))^1); polcoeff(A, n)
(PARI) {a(n)=local(A=1+x); if(n==0, 1, for(i=1, n,
A=1+x*exp(sum(k=1, n-1, sum(j=0, k, polcoeff(A^k+x*O(x^j), j))*x^k/k)+x*O(x^n))));
polcoeff(A+x*O(x^n), n)}
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