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EXAMPLE
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G.f.: A(x) = 1 + x - 3*x^2 + 14*x^3 - 85*x^4 + 504*x^5 - 4424*x^6 +...
Coefficients of x^k in the powers A(x)^(n^2+n+1) of g.f. A(x) begin:
n=0: [1, 1, -3, 14, -85, 504, -4424, 6796, ...];
n=1: [1, 3, -6, 25, -153, 819, -8664, -18360, ...];
n=2: [1, 7, 0, 7, -98, 210, -10122, -141525, ...];
n=3: [1,13, 39, 0, -78, -819, -15483, -380952, ...];
n=4: [1,21, 147, 364, 0, -2457, -35805, -821916, ...];
n=5: [1,31, 372, 2139, 5580, 0, -91698, -1792947, ...];
n=6: [1,43, 774, 7525, 42097, 125517, 0, -4097298, ...];
n=7: [1,57, 1425, 20482, 185877, 1089270, 3791298, 0, ...];
n=8: [1,73, 2409, 47450, 619697, 5619978, 35621518, 144591976, 0, ...]; ...
where the coefficients of x^n in A(x)^(n^2+n+1) all equal zero for n>1.
ODD TERMS:
For n>0, a(n) appears to be odd only when n is a power of 2:
a(1) = 1;
a(2) = -3;
a(4) = -85;
a(8) = -878157;
a(16) = -111112550557260635229;
a(32) = -886203693344229341179357569730608605545213045330679133; ...
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