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A373314
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Expansion of g.f. A(x) satisfying A(A(x)) - 4*A(A(A(x)))^2 = x.
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0
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1, 2, 20, 304, 5728, 123680, 2942016, 75356544, 2048446208, 58517294080, 1744472116224, 53991571224576, 1728111953805312, 57027260271980544, 1935586663121272832, 67440373642584637440, 2408328339125296824320, 88029604711420113190912, 3289877540493975587913728, 125591805077248068782129152
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = A(A(x)) - 4*A(A(A(x)))^2.
(2) x = A( A(x) - 4*A(A(x))^2 ).
(3) x = A(A( x - 4*A(x)^2 )).
(4) A(x) = A(A(A( x - 4*A(x)^2 ))).
(5) A(x) = A(A(A(x))) - 4*A(A(A(A(x))))^2.
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EXAMPLE
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G.f.: A(x) = x + 2*x^2 + 20*x^3 + 304*x^4 + 5728*x^5 + 123680*x^6 + 2942016*x^7 + 75356544*x^8 + 2048446208*x^9 + 58517294080*x^10 + ...
where A(A(x)) - 4*A(A(A(x)))^2 = x.
RELATED SERIES.
A(A(x)) = x + 4*x^2 + 48*x^3 + 816*x^4 + 16704*x^5 + 385600*x^6 + 9705728*x^7 + 261167104*x^8 + ...
A(A(A(x))) = x + 6*x^2 + 84*x^3 + 1584*x^4 + 35168*x^5 + 869152*x^6 + 23222336*x^7 + 659257728*x^8 + ...
A(A(A(x)))^2 = x^2 + 12*x^3 + 204*x^4 + 4176*x^5 + 96400*x^6 + 2426432*x^7 + 65291776*x^8 + ...
Let B(x) be the series reversion of A(A(x)), B( A(A(x)) ) = x, then
B(x) = x - 4*A(x)^2 = x - 4*x^2 - 16*x^3 - 176*x^4 - 2752*x^5 - 52288*x^6 - 1129728*x^7 - 26801152*x^8 - ...
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PROG
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(PARI) /* Using x = A(A(x)) - 4*A(A(A(x)))^2 */
{a(n) = my(A = [0, 1], A1, A2, A3); for(i=1, n, A = concat(A, 0); A1 = Ser(A);
A2 = subst(A1, x, A1); A3 = subst(A1, x, A2);
A[#A] = (1/2)*polcoeff(x - A2 + 4*A3^2, #A-1)); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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