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 A120009 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. 8
 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; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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. LINKS FORMULA 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). 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, Nov 21 2006 D-finite with recurrence: (n+2)*a(n) +(-7*n-2)*a(n-1) +6*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020, corrected Feb 16 2020 EXAMPLE 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 +... PROG (PARI) a(n)=binomial(2*n, n)/(n+1)-4*binomial(2*n-1, n-2)/(n+2) CROSSREFS Cf. A120010 (composition transpose), A000108 (Catalan). cf. A003517 (|a(n+1)|-|a(n)|). From Olivier Gérard, Oct 11 2012 Sequence in context: A263479 A073375 A089097 * A074087 A297592 A255613 Adjacent sequences:  A120006 A120007 A120008 * A120010 A120011 A120012 KEYWORD sign AUTHOR Paul D. Hanna, Jun 03 2006 STATUS approved

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Last modified May 27 16:48 EDT 2022. Contains 354110 sequences. (Running on oeis4.)