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A371708
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Expansion of g.f. A(x) satisfying A( x*A(x - x^2) ) = x^2.
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3
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1, 1, 1, 2, 6, 19, 60, 193, 636, 2141, 7331, 25451, 89385, 317036, 1134100, 4087104, 14825482, 54088470, 198348985, 730723956, 2703194553, 10037648254, 37399878530, 139785998185, 523962161491, 1969154471389, 7418488063284, 28010998254007, 105986233046356, 401804972780552
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OFFSET
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1,4
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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, along with its series reversion R(x), satisfy the following formulas.
(1) A( x*A(x - x^2) ) = x^2.
(2) A(x - x^2) = R(x^2)/x.
(3) (R(x) - R(-x))^2 + 2*(R(x) + R(-x)) = 0.
(4) R(x) = R(-x) - 1 + sqrt(1 - 4*R(-x)).
(5) A(x) = -A( x - 1 + sqrt(1 - 4*x) ).
(6) A(x) = -A(x - 2*C(x)) where C(x) = -C(x - 2*C(x)) is a g.f. of the Catalan numbers (A000108).
(7) A( A(x)*C(x) ) = C(x)^2 where C(x) = (1 - sqrt(1 - 4*x))/2 is a g.f. of the Catalan numbers (A000108).
a(n) ~ c * 4^n / n^(3/2), where c = 0.0517683007874758928168667... - Vaclav Kotesovec, Apr 24 2024
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 60*x^7 + 193*x^8 + 636*x^9 + 2141*x^10 + 7331*x^11 + 25451*x^12 + 89385*x^13 + 317036*x^14 + ...
where A( x*A(x - x^2) ) = x^2.
RELATED SERIES.
Let R(x) be the series reversion of A(x), A(R(x)) = x, which begins
R(x) = x - x^2 + x^3 - 2*x^4 + 2*x^5 - 5*x^6 + 6*x^7 - 16*x^8 + 23*x^9 - 62*x^10 + 100*x^11 - 270*x^12 + 463*x^13 - 1254*x^14 + 2224*x^15 - 6050*x^16 + ...
then R( R(x^2)/x ) = x - x^2.
Also, the bisections B1 and B2 of R(x) are
B1 = (R(x) - R(-x))/2 = x + x^3 + 2*x^5 + 6*x^7 + 23*x^9 + 100*x^11 + 463*x^13 + 2224*x^15 + 10963*x^17 + ...
B2 = (R(x) + R(-x))/2 = -x^2 - 2*x^4 - 5*x^6 - 16*x^8 - 62*x^10 - 270*x^12 - 1254*x^14 - 6050*x^16 + ...
and satisfy B1^2 + B2 = 0 and A(x*B1) = B1^2.
SPECIFIC VALUES.
A( A(1/4) / 2 ) = 1/4 where
A(1/4) = 0.39241307250698647662923990494867613212061604622566765...
A( A(2/9) / 3 ) = 1/9 where
A(2/9) = 0.29957319341272312632777466712131772539171747971866175...
A( A(3/16) / 4 ) = 1/16 where
A(3/16) = 0.2352360051274118086289466324430753987734355106832392...
A( A(4/25) / 5 ) = 1/25 where
A(4/25) = 0.1922953260179964363449115205476634347705922222443464...
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PROG
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(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( x^2 - subst(Ser(A), x, x*subst(Ser(A), x, x - x^2) ), #A)); A[n+1]}
for(n=1, 35, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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