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A271958
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G.f. A(x) satisfies: A( x*A(x) - 2*x*A(x)^2 ) = x^2.
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2
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1, 2, 6, 32, 178, 1040, 6340, 40000, 258498, 1702640, 11390428, 77184064, 528665660, 3654285120, 25458887576, 178585843712, 1260264395114, 8940927869584, 63732097097436, 456221687622976, 3278343806166228, 23639492652661312, 170998398398401832, 1240505401469155840, 9023083581252148332, 65791560086084269248, 480799272958960336552, 3520985178220566410496, 25834931848716275409960
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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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Let B(x) be the series reversion of A(x) so that A(B(x)) = x, then
(1) B(x) = B(B(x)^2) / (x - 2*x^2).
(2) B(x^2) = x*A(x) - 2*x*A(x)^2.
(3) A(x) - 2*A(x)^2 is an odd function.
(4) A(x) = (1 - sqrt(1 - 8*B(x^2)/x)) / 4.
(5) A(x) = C( 2*B(x^2)/x ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers A000108.
a(n) ~ c * d^n / n^(3/2), where d = 7.73673466070656248242789... and c = 0.0681159017783868428596... . - Vaclav Kotesovec, Apr 18 2016
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EXAMPLE
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G.f.: A(x) = x + 2*x^2 + 6*x^3 + 32*x^4 + 178*x^5 + 1040*x^6 + 6340*x^7 + 40000*x^8 + 258498*x^9 + 1702640*x^10 + 11390428*x^11 + 77184064*x^12 +...
such that A( x*A(x) - 2*x*A(x)^2 ) = x^2
where
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 88*x^5 + 520*x^6 + 3176*x^7 + 20000*x^8 + 129232*x^9 + 851320*x^10 + 5695272*x^11 + 38592032*x^12 +...
and
A(x) - 2*A(x)^2 = x - 2*x^3 + 2*x^5 - 12*x^7 + 34*x^9 - 116*x^11 + 476*x^13 - 1864*x^15 + 7850*x^17 - 33588*x^19 + 144756*x^21 - 637048*x^23 + 2823468*x^25 +...
which is an odd function.
Compare with B(x), the series reversion of A(x), A(B(x)) = x:
B(x) = x - 2*x^2 + 2*x^3 - 12*x^4 + 34*x^5 - 116*x^6 + 476*x^7 - 1864*x^8 + 7850*x^9 - 33588*x^10 + 144756*x^11 - 637048*x^12 + 2823468*x^13 +...
which satisfies: B(B(x)^2) = (x - 2*x^2)*B(x).
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PROG
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(PARI) {a(n) = my(A=x); for(i=1, #binary(n), A = (1 - sqrt(1 - 8*subst(serreverse(A +x*O(x^n)), x, x^2)/x) )/4 ); 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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