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Search: id:A120009
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| A120009 |
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G.f.: A(x) = (x-x^2) o x/(1-x) o (1-sqrt(1-4*x))/2, a composition of functions involving the Catalan function and its inverse. |
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8
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| 1, 1, 1, 0, -6, -33, -143, -572, -2210, -8398, -31654, -118864, -445740, -1671525, -6273135, -23571780, -88704330, -334347090, -1262330850, -4773905760, -18083762580, -68611922730, -260725306374, -992233959480, -3781513867796, -14431491699548, -55147299002348
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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The n-th self-composition of A(x) is: (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2. See A120010 for the transpose of the composition of the same functions.
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FORMULA
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G.f.: A(x) = ((1-3*x)*sqrt(1-4*x) - (1-x)*(1-4*x))/(2*x^2) = x*C(x)^2 - x^2*C(x)^4 where C(x) is the Catalan function (A000108). Formula: a(n) = C(2*n,n)/(n+1) - 4*C(2*n-1,n-2)/(n+2).
a(n) = 3*CatalanNumber[n] - CatalanNumber[n+1]. - David Callan (callan(AT)stat.wisc.edu), Nov 21 2006
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EXAMPLE
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A(x) = x + x^2 + x^3 - 6*x^5 - 33*x^6 - 143*x^7 - 572*x^8 - 2210*x^9 +...
A(x) = x*C(x)^2 - x^2*C(x)^4 where C(x) is Catalan function so that:
x*C(x)^2 = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...
x^2*C(x)^4 = x^2 + 4*x^3 + 14*x^4 + 48*x^5 + 165*x^6 + 572*x^7 +...
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PROGRAM
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(PARI) a(n)=binomial(2*n, n)/(n+1)-4*binomial(2*n-1, n-2)/(n+2)
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CROSSREFS
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Cf. A120010 (composition transpose), A000108 (Catalan).
Sequence in context: A082106 A073375 A089097 this_sequence A074087 A022730 A099432
Adjacent sequences: A120006 A120007 A120008 this_sequence A120010 A120011 A120012
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KEYWORD
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sign
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jun 03 2006
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