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A373311
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Expansion of g.f. A(x) satisfying A( A(x^2) + C(x) ) = x, where C(x) = x + C(x)^2 is the Catalan function (A000108).
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2
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1, -2, 6, -23, 90, -370, 1568, -6802, 30032, -134422, 608136, -2774480, 12741844, -58820272, 272617696, -1267318854, 5904152560, -27545442786, 128610298720, -600579852616, 2803385644716, -13072680693872, 60864695165024, -282763138097520, 1309945271204312, -6047009494466692
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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, along with the Catalan function C(x), satisfies the following formulas.
(1) A( A(x^2) + C(x) ) = x.
(2) A(x^2) = B(x) - C(x), where B(A(x)) = x (cf. A373310).
(3) A( x + A(x^2*(1-x)^2) ) = x - x^2.
(4) A( x^2*(1 - x)^2 ) = B(x-x^2) - x, where B(A(x)) = x.
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EXAMPLE
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G.f.: A(x) = x - 2*x^2 + 6*x^3 - 23*x^4 + 90*x^5 - 370*x^6 + 1568*x^7 - 6802*x^8 + 30032*x^9 - 134422*x^10 + 608136*x^11 - 2774480*x^12 + ...
where A( A(x^2) + C(x) ) = x and C(x) = x + C(x)^2.
RELATED SERIES.
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n)*x^n + ...
where C(x) = (1 - sqrt(1 - 4*x))/2.
Let B(x) be the series reversion of A(x), B(A(x)) = x, then
B(x) = x + 2*x^2 + 2*x^3 + 3*x^4 + 14*x^5 + 48*x^6 + 132*x^7 + 406*x^8 + 1430*x^9 + 4952*x^10 + ... + A373310(n)*x^n + ...
where B(x) = A(x^2) + C(x).
SPECIFIC VALUES.
A(1/8^2) = 0.0151583147955015735432348644518273813510395283898972054...
where A( A(1/8^2) + (1 - sqrt(1/2))/2 ) = 1/8.
A(1/10^2) = 0.009805778645028545940916921405996691003884354282840359...
where A( A(1/10^2) + (1 - sqrt(3/5))/2 ) = 1/10.
A(1/6) = 0.12804766097193434952321787018897962946429947564936619827...
A(1/7) = 0.11334295578613203758002178169147530906130640897889608109...
A(1/8) = 0.10167460278631364378329951659088472698712642799619893229...
A(1/16) = 0.0558697677874474568531344354179668140114433992762805769...
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PROG
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(PARI) {a(n) = my(A = x +x*O(x^n), C = serreverse(x-x^2 +x*O(x^n)));
for(i=1, #binary(n), A = serreverse(subst(A, x, x^2) + C)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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