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A292928
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G.f. A(x) satisfies: A( x^2*A(x) - A(x)^4 ) = x^3.
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3
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1, 1, 4, 21, 136, 941, 6862, 51998, 405256, 3228208, 26167124, 215135798, 1789732224, 15037735467, 127429439672, 1087804527493, 9345903237508, 80751170130654, 701224835836174, 6116663059682404, 53570252214591940, 470886419343198751, 4152853930912625384, 36735754728365394280, 325861321839388361170, 2897885772356354420332
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^(3/2), where d = 9.4477112726509666047269249019968778065... and c = 0.01614622962039842631454... - Vaclav Kotesovec, Oct 10 2017
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EXAMPLE
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G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 136*x^5 + 941*x^6 + 6862*x^7 + 51998*x^8 + 405256*x^9 + 3228208*x^10 + 26167124*x^11 + 215135798*x^12 + 1789732224*x^13 + 15037735467*x^14 + 127429439672*x^15 + 1087804527493*x^16 +...
such that A( x^2*A(x) - A(x)^4 ) = x^3.
RELATED SERIES.
x^2*A(x) - A(x)^4 = x^3 - x^6 - 2*x^9 - 6*x^12 - 32*x^15 - 143*x^18 - 718*x^21 - 3866*x^24 - 21054*x^27 - 118409*x^30 +...
Let B(x) be the series reversion of A(x), so that B(A(x)) = x, then
B(x) = x - x^2 - 2*x^3 - 6*x^4 - 32*x^5 - 143*x^6 - 718*x^7 - 3866*x^8 - 21054*x^9 - 118409*x^10 - 679560*x^11 - 3955972*x^12 - 23343650*x^13 - 139235773*x^14 - 838057508*x^15 - 5084422207*x^16 +...
then x^2*A(x) - A(x)^4 = B(x^3).
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PROG
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(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -Vec(subst(F, x, x^2*F - F^4))[#A] ); polcoeff(A, n)}
for(n=1, 40, 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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