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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 217*x^4 + 2814*x^5 + 42510*x^6 +...
Related expansions.
A(x)^3 = 1 + 3*x + 12*x^2 + 82*x^3 + 813*x^4 + 10212*x^5 + 150699*x^6 +...
A(A(x)-1) = 1 + x + 6*x^2 + 60*x^3 + 776*x^4 + 11802*x^5 + 201465*x^6 +...
A(A(x)-1)^3 = 1 + 3*x + 21*x^2 + 217*x^3 + 2814*x^4 + 42510*x^5 +...
x/A(x)^3 = x - 3*x^2 - 3*x^3 - 37*x^4 - 420*x^5 - 5823*x^6 -...
Series_Reversion(x/A(x)^3) = x + 3*x^2 + 21*x^3 + 217*x^4 + 2814*x^5 + 42510*x^6 +...
To illustrate the formula a(n) = [x^(n-1)] 3*A(x)^(3*n)/(3*n),
form a table of coefficients in A(x)^(3*n) as follows:
A^3: [(1), 3, 12, 82, 813, 10212, 150699, 2503233, ...];
A^6: [1, (6), 33, 236, 2262, 27270, 388906, 6289080, ...];
A^9: [1, 9, (63), 489, 4671, 54684, 756012, 11904813, ...];
A^12: [1, 12, 102, (868), 8445, 97260, 1310040, 20112516, ...];
A^15: [1, 15, 150, 1400, (14070), 161343, 2130505, 31961175, ...];
A^18: [1, 18, 207, 2112, 22113, (255060), 3324003, 48876264, ...];
A^21: [1, 21, 273, 3031, 33222, 388563, (5030529), 72769014, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[3/3*(1), 3/6*(6), 3/9*(63), 3/12*(868), 3/15*(14070), 3/18*(255060), ...].
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PROG
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(PARI) {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[ #A]=-Vec(subst(Ser(A), x, x/Ser(A)^3))[ #A]); A[n+1]}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*subst(A^3, x, A-1+x*O(x^n))); polcoeff(A, n)}
(PARI) /* This sequence is generated when k=3, m=0: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=3, m=0)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0, 25, print1(a(n), ", "))
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