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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 126*x^4 + 966*x^5 + 7863*x^6 + 66696*x^7 + 583111*x^8 + 5217513*x^9 + 47547405*x^10 + ...
such that
A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).
Also,
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
RELATED TABLE.
The table of coefficients in A(x)^n begins:
n=1: [1, 1, 3, 18, 126, 966, 7863, 66696, ...];
n=2: [1, 2, 7, 42, 297, 2292, 18738, 159450, ...];
n=3: [1, 3, 12, 73, 522, 4059, 33354, 284886, ...];
n=4: [1, 4, 18, 112, 811, 6360, 52566, 450888, ...];
n=5: [1, 5, 25, 160, 1175, 9301, 77370, 666780, ...];
n=6: [1, 6, 33, 218, 1626, 13002, 108919, 943524, ...];
n=7: [1, 7, 42, 287, 2177, 17598, 148540, 1293937, ...];
n=8: [1, 8, 52, 368, 2842, 23240, 197752, 1732928, ...];
n=9: [1, 9, 63, 462, 3636, 30096, 258285, 2277756, ...];
...
from which one can verify the formulas involving powers of A(x).
RELATED SERIES.
Let G(x) = 1 + Series_Reversion( x/(1 + x*(1+x)^2 + x*(1+x)^5) )
where
G(x) = 1 + x + 2*x^2 + 11*x^3 + 61*x^4 + 380*x^5 + 2502*x^6 + 17163*x^7 + 121312*x^8 + 877370*x^9 + 6461765*x^10 + ...
then
A(x) = G(x*A(x)),
and so
A(x) = (1/x) * Series_Reversion( x/G(x) );
thus,
x*A(x) = (A(x) - 1) / (1 + (A(x) - 1)*(A(x)^2 + A(x)^5) )
which is equivalent to
A(x) = 1 + x*(A(x) - A(x)^3 + A(x)^4 - A(x)^6 + A(x)^7).
TERMS MODULO 3.
It appears that a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).
The residues of a(7*k + 1) modulo 3, for k >= 0, begin
a(7*k + 1) (mod 3) = [1, 1, 1, 1, 0, 2, 1, 0, 0, 1, 1, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, ...].
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