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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4312*x^5 + 28704*x^6 +...
Given g.f. A(x), let q = x*A(x), then by a q-series identity:
A(x) = 1 + 4*q/(1+q^2) + 4*q^2/(1+q^4) + 4*q^3/(1+q^6) + 4*q^4/(1+q^8) +...
A(x) = (1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 +...)^2.
...
Illustrate a(n) = [x^n] theta_3(x)^(2*n+2) / (n+1) by the following table of coefficients in powers theta_3(x)^(2*n+2) for n>=0:
n=0: [(1), 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0,...];
n=1: [1, (8), 24, 32, 24, 48, 96, 64, 24, 104, 144, 96, 96, 112,...];
n=2: [1, 12, (60), 160, 252, 312, 544, 960, 1020, 876, 1560, 2400,...];
n=3: [1, 16, 112, (448), 1136, 2016, 3136, 5504, 9328, 12112,...];
n=4: [1, 20, 180, 960, (3380), 8424, 16320, 28800, 52020, 88660,...];
n=5: [1, 24, 264, 1760, 7944, (25872), 64416, 133056, 253704,...];
n=6: [1, 28, 364, 2912, 16044, 64792, (200928), 503360, ...];
n=7: [1, 32, 480, 4480, 29152, 140736, 525952, (1580800), ...]; ...
where the coefficients in parenthesis form the initial terms of this sequence:
A = [1/1, 8/2, 60/3, 448/4, 3380/5, 25872/6, 200928/7, 1580800/8, ...].
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+4*sum(m=1, n, (x*A)^m/(1+(x*A+x*O(x^n))^(2*m)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+2*sum(m=1, sqrtint(n+1), (x*A+x*O(x^n))^(m^2)))^2); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1-(-x)^m*A^m)/(1+(-x)^m*A^m +x*O(x^n)))^2); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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