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A180254
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E.g.f. A(x) satisfies: A'(x) = (1 - sqrt(1-4*A(x))) / (2*A(x)).
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2
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1, 1, 5, 47, 655, 12145, 281645, 7848575, 255581095, 9527142625, 400145871125, 18700514207375, 962618343061375, 54121350164907025, 3300097924150152125, 216926019220561025375, 15292057853288541700375, 1150861860384703732890625, 92099021113080260667513125
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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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E.g.f. A(x) satisfies: A'(x) = 1/(1 - A(x)*A'(x)) = 1 + A(x)*A'(x)^2. - Paul D. Hanna, Jul 24 2012
E.g.f. A(x) satisfies: A'(x) = Sum_{n>=0} A000108(n)*A(x)^n where A000108(n) = binomial(2*n,n)/(n+1) forms the Catalan numbers. - Paul D. Hanna, Jul 24 2012
E.g.f.: Series_Reversion( Integral( dx/C(x) ) ) where C(x) = 1 + x*C(x)^2 is the Catalan function of A000108. - Paul D. Hanna, Dec 09 2013
Recurrence: 5*a(n) = 3*(18*n-35)*a(n-1) - 16*(3*n-8)*(3*n-7)*a(n-2). - Vaclav Kotesovec, Feb 19 2014
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EXAMPLE
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E.g.f.: A(x) = x + x^2/2! + 5*x^3/3! + 47*x^4/4! + 655*x^5/5! +...
where, by definition, A(x)*A'(x) = (1 - sqrt(1-4*A(x)))/2:
A(x)*A'(x) = x + 3*x^2/2! + 23*x^3/3! + 285*x^4/4! + 4885*x^5/5! +...
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MATHEMATICA
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terms = 20; B[_] = 1; Do[B[x_] = (1 - Sqrt[1 - 4*Integrate[B[x], x]])/ (2*Integrate[B[x], x]) + O[x]^terms // Normal, terms];
A[x_] = Integrate[B[x], x];
a[1] = a[2] = 1; a[n_] := a[n] = (1/5)*(-(144*n^2*a[n-2]) + 720*n*a[n-2] + 54*n*a[n-1] - 896*a[n-2] - 105*a[n-1]); Array[a, 20] (* Jean-François Alcover, Apr 21 2016 *)
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PROG
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(PARI) a(n)=local(A=x); for(i=1, n, A=intformal(1/(1-A*A'+x*O(x^n)))); n!*polcoeff(A, n) \\ Paul D. Hanna, Jul 24 2012
for(n=1, 20, print1(a(n), ", "))
(PARI) a(n)=local(C=(1-sqrt(1-4*x+x^2*O(x^n)))/(2*x), A); n!*polcoeff(serreverse(intformal(1/C)), 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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