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A373315
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Expansion of g.f. A(x) satisfying A(A(x))^2 - 8*A(A(A(x)))^3 = x^2.
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2
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1, 2, 28, 592, 15360, 452864, 14607168, 504327808, 18383015936, 700853891584, 27763279234048, 1137180367163392, 47984104148398080, 2079829668276404224, 92391710881374027776, 4198734066925714538496, 194910502907612385968128, 9230979805443695622225920, 445562401524920751310503936
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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^2 = A(A(x))^2 - 8*A(A(A(x)))^3.
(2) x = A( sqrt( A(x)^2 - 8*A(A(x))^3 ) ).
(3) x = A(A( sqrt( x^2 - 8*A(x)^3 ) )).
(4) A(x) = A(A(A( sqrt( x^2 - 8*A(x)^3 ) ))).
(5) A(x)^2 = A(A(A(x)))^2 - 8*A(A(A(A(x))))^3.
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EXAMPLE
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G.f.: A(x) = x + 2*x^2 + 28*x^3 + 592*x^4 + 15360*x^5 + 452864*x^6 + 14607168*x^7 + 504327808*x^8 + 18383015936*x^9 + 700853891584*x^10 + ...
where A(A(x))^2 - 8*A(A(A(x)))^3 = x^2.
RELATED SERIES.
A(A(x)) = x + 4*x^2 + 64*x^3 + 1472*x^4 + 40736*x^5 + 1266944*x^6 + 42797184*x^7 + 1539469824*x^8 + ...
A(A(A(x))) = x + 6*x^2 + 108*x^3 + 2688*x^4 + 79200*x^5 + 2598528*x^6 + 92039232*x^7 + 3456376192*x^8 + ...
A(A(x))^2 = x^2 + 8*x^3 + 144*x^4 + 3456*x^5 + 97344*x^6 + 3048192*x^7 + 103110912*x^8 + 3703412736*x^9 + ...
A(A(A(x)))^3 = x^3 + 18*x^4 + 432*x^5 + 12168*x^6 + 381024*x^7 + 12888864*x^8 + 462926592*x^9 + ...
Let B(x) be the series reversion of A(A(x)), B( A(A(x)) ) = x, then
B(x) = sqrt( x^2 - 8*A(x)^3 ) = x - 4*x^2 - 32*x^3 - 512*x^4 - 11040*x^5 - 284032*x^6 - 8237184*x^7 - 260861952*x^8 - ...
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PROG
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(PARI) /* Using x^2 = A(A(x))^2 - 8*A(A(A(x)))^3 */
{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] = (1/4)*polcoeff(x^2 - A2^2 + 8*A3^3, #A)); 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
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AUTHOR
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STATUS
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approved
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