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.
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
Offset: 1
Examples
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 + ...
Crossrefs
Cf. A003517 (|a(n+1)|-|a(n)|). - Olivier Gérard, Oct 11 2012
Programs
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Magma
[3*Catalan(n) - Catalan(n+1): n in [1..30]]; // Vincenzo Librandi, Jan 02 2025
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Mathematica
f[n_]:=3*CatalanNumber[n] -CatalanNumber[n+1];Array[f,30,1] (* Vincenzo Librandi, Jan 02 2025 *)
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PARI
a(n)=binomial(2*n,n)/(n+1)-4*binomial(2*n-1,n-2)/(n+2)
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*Catalan(n) - Catalan(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
From Peter Bala, Feb 02 2024: (Start)
a(n) = 3*(-1)^n*Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*(2*k + 2)!/((k + 3)!*k!).
G.f.: x/(1 - 4*x)*c(-x/(1 - 4*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
E.g.f.: exp(2*x)*(2*BesselI(0, 2*x) - 3*BesselI(1, 2*x) + BesselI(2, 2*x)) - 2. - Stefano Spezia, Dec 31 2024
Comments