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A370146
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Expansion of x / Series_Reversion( x/(1 + 3*x - 6*x^2 - 8*x^3)^(1/3) ).
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3
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1, -1, 3, 0, 0, 0, -9, 0, 27, 0, 0, 0, -324, 0, 1215, 0, 0, 0, -18711, 0, 75816, 0, 0, 0, -1301265, 0, 5484996, 0, 0, 0, -100048689, 0, 431943435, 0, 0, 0, -8192222064, 0, 35942240565, 0, 0, 0, -700434986472, 0, 3108770417700, 0, 0, 0, -61805774132388, 0, 276711654879477
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OFFSET
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0,3
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COMMENTS
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The cube root of F(x) = (1 + 3*x - 6*x^2 - 8*x^3) = (1 + x)*(1 - 2*x)*(1 + 4*x) is an integer series because F(x) == (1+x)^3 (mod 9).
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = x / Series_Reversion( x/(1 + 3*x - 6*x^2 - 8*x^3)^(1/3) ).
(2) A(x) = 1 / B(x/A(x)) where B(x) = 1 / A(x/B(x)) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) equals the g.f. of A370145.
(3) A(x) = 1/C(27*x^6)^(1/3) + 3*x^2*C(27*x^6)^(1/3) - x, where C(x) = 1 + x*C(x)^2 = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).
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EXAMPLE
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G.f.: A(x) = = 1 - x + 3*x^2 - 9*x^6 + 27*x^8 - 324*x^12 + 1215*x^14 - 18711*x^18 + 75816*x^20 - 1301265*x^24 + 5484996*x^26 - 100048689*x^30 + ...
RELATED SERIES.
If A(x) = 1/B(x/A(x)) then B(x) = (1 + 3*x - 6*x^2 - 8*x^3)^(1/3) begins
B(x) = 1 + x - 3*x^2 + 3*x^3 - 12*x^4 + 30*x^5 - 102*x^6 + 318*x^7 - 1083*x^8 + 3657*x^9 - 12747*x^10 + 44715*x^11 + ... + A370145(n)*x^n + ...
A(x) = 1/D(x^6) + 3*x^2*D(x^6) - x, where
1/D(x) = 1 - 9*x - 324*x^2 - 18711*x^3 - 1301265*x^4 - 100048689*x^5 - 8192222064*x^6 - 700434986472*x^7 + ...
and D(x) = ( (1-sqrt(1-108*x))/(54*x) )^(1/3) begins
D(x) = 1 + 9*x + 405*x^2 + 25272*x^3 + 1828332*x^4 + 143981145*x^5 + 11980746855*x^6 + 1036256805900*x^7 + ... + 3^n*A008931(n)*x^n + ...
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PROG
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(PARI) {a(n) = polcoeff( x/serreverse( x/(1 + 3*x - 6*x^2 - 8*x^3 +x*O(x^n))^(1/3) ), n)}
for(n=0, 50, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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