OFFSET
0,3
COMMENTS
More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), |b|>0, |c|>0, then Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1112
Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
Paul Barry and Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.
FORMULA
G.f.: sqrt( (1-2*x - sqrt(1-4*x-32*x^2))/2 )/(3*x).
G.f.: (1/x)*Series_Reversion(x-x^2 - 4*x^3*Sum_{n>=0} A000108(n)*3^n*x^(2*n) ).
G.f. satisfies: A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) is the g.f. of A200376: G(x) = 1/sqrt(1-10*x^2 + x^4/(1-8*x^2)) + x/(1-9*x^2).
n*(n+1)*a(n) -2*n*(2*n-1)*a(n-1) -8*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 17 2011
a(n) = binomial(2*n,n)/(n+1) * (2^(n+1) + (-1)^n)/3.
From Peter Bala, Aug 17 2021: (Start)
G.f.: A(x) = (sqrt(1 + 4*x) - sqrt(1 - 8*x))/(6*x).
A(x) = 1/sqrt(1 + 4*x)*c( 3*x/(1 + 4*x) ), where c(x) = (1 - sqrt(1- 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Cf. A151374.
In general, [x^n] ( 1/sqrt(1 + 4*x)*c( k*x/(1 + 4*x) ) ) = Catalan(n)*((k-1)^(n+1) + (-1)^(n+1))/k.
A(x) = 1/sqrt(1 - 8*x)*c( -3*x/(1 - 8*x) ). (End)
EXAMPLE
G.f.: A(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 + 429*85*x^7 + 1430*171*x^8 +...+ A000108(n)*A001045(n)*x^n +...
The g.f. of the Jacobsthal sequence A001045, F(x) = 1/(1-x-2*x^2), begins:
F(x) = 1 + x + 3*x^2 + 5*x^3 + 11*x^4 + 21*x^5 + 43*x^6 + 85*x^7 + 171*x^8 +...
The g.f. of A200376, where G(x) = A(x/G(x)), begins:
G(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...
in which the odd-indexed coefficients are powers of 9.
MATHEMATICA
Array[CatalanNumber[# - 1] (2^# - (-1)^#)/3 &, 25] (* Michael De Vlieger, Apr 24 2018 *)
PROG
(PARI) {a(n)=binomial(2*n, n)/(n+1)*(2^(n+1)+(-1)^n)/3}
(PARI) {a(n)=polcoeff(sqrt((1-2*x - sqrt(1-4*x-32*x^2+O(x^(n+3))))/2)/(3*x), n)}
(PARI) {a(n)=polcoeff((1/x)*serreverse(x-x^2 - 4*x^3*sum(m=0, n\2, binomial(2*m, m)/(m+1)*3^m*x^(2*m))+x^3*O(x^n)), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 16 2011
EXTENSIONS
Typo in Name corrected by Peter Bala, Aug 17 2021
STATUS
approved