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EXAMPLE
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G.f.: A(x) = 1 + 6*x - 36*x^2 + 360*x^3 - 4320*x^4 + 57024*x^5 - 816480*x^6 + 12737088*x^7 - 216950400*x^8 + 3985853184*x^9 + ...
The table of coefficients of x^k in A(x)^(n*(n+1)/2), for k>=0, begins:
n=1: [1, 6, -36, 360, -4320, 57024, -816480, 12737088, -216950400, ...];
n=2: [1, 18, 0, 0, 0, 0, -54432, 2659392, -85660416, 2267481600, -...];
n=3: [1, 36, 324, 0, 0, 0, -108864, 3359232, -75582720, 1451188224, ...];
n=4: [1, 60, 1260, 10080, 15120, -36288, 0, 0, 5598720, -245099520, ...];
n=5: [1, 90, 3240, 58320, 524880, 1889568, -272160, -6298560, 0, 0, 0, 0, ...]; ...
in which both coefficients of x^(2*n-2) and x^(2*n-1) in A(x)^(n^2) equal zero for n > 1.
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PROG
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(PARI) {a(n) = my(A=[1, 6], P); for(i=1, n, A=concat(A, 0); P = (#A+1)\2;
A[#A] = -polcoeff( Ser(A)^(P*(P+1)/2)/(P*(P+1)/2), #A-1) ); H=A; A[n+1]}
for(n=0, 20, print1(a(n), ", "))
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