OFFSET
0,3
COMMENTS
Radius of convergence: r = (sqrt(5)-1)/8; A(r) = sqrt(2+2/sqrt(5)). 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.
a(n) is also the number of nonnesting permutations of {1,1,2,2,...,n,n} that avoid the patterns 1223, 1332, 2113, or the patterns 1123, 1132, 2133. - Amya Luo, Dec 11 2024
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
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.
Sergi Elizalde and Amya Luo, Pattern avoidance in nonnesting permutations, arXiv:2412.00336 [math.CO], 2024.
FORMULA
G.f.: A(x) = sqrt( (1-2*x - sqrt(1-4*x-16*x^2))/10 )/x.
G.f. satisfies: A(x) = sqrt( 1 + 2*x*A(x)^2 + 5*x^2*A(x)^4 ).
a(n) == 1 (mod 2) iff n = 2^k - 1 for k>=0.
n*(n+1)*a(n) -2*n*(2*n-1)*a(n-1) -4*(2*n-1)*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 17 2018
Sum_{n>=0} a(n)/8^n = 2*sqrt(2/5). - Amiram Eldar, May 06 2023
EXAMPLE
Sequence has the factored form: {1*1, 1*1, 2*2, 3*5, 5*14, 8*42, 13*132, 21*429, ...}.
MATHEMATICA
With[{nn=30}, Times@@@Thread[{Fibonacci[Range[nn]], CatalanNumber[ Range[ 0, nn-1]]}]] (* Harvey P. Dale, Nov 14 2011 *)
PROG
(PARI) {a(n)=local(X=x+O(x^(n+3)), A); A = sqrt( (1-2*x - sqrt(1-4*X-16*x^2)) / (10*x^2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=binomial(2*n, n)/(n+1)*round(((1+sqrt(5))^(n+1)-(1-sqrt(5))^(n+1))/(2^(n+1)*sqrt(5)))}
(Magma) [Fibonacci(n+1)*Catalan(n): n in [0..40]]; // G. C. Greubel, Jul 31 2024
(SageMath) [fibonacci(n+1)*catalan_number(n) for n in range(41)] # G. C. Greubel, Jul 31 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 09 2004
STATUS
approved