OFFSET
0,2
COMMENTS
The Hankel transform of this sequence is 7^n = [1, 7, 49, 343, 2401, ...] . The Hankel transform of the aerated sequence with g.f. 1/(1 - 7*x^2*c(x^2)) is also 7^n.
Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from c(x) = 1/(1 - x*c(x)) == 1/(1 - 7*x*c(x)) (mod 2). - Peter Bala, Jul 24 2016
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(0) = 1, a(n) = (49*a(n-1) - 7*A000108(n-1))/6 for n >= 1.
a(n) = Sum_{k = 0..n} A106566(n,k)*7^k.
a(n) = Sum_{k = 0..n} A039599(n,k)*6^k.
a(n) ~ 5 * 7^(2*n) / 6^(n+1). - Vaclav Kotesovec, Nov 29 2021
MATHEMATICA
CoefficientList[Series[2/(-5+7*Sqrt[1-4*x]), {x, 0, 30}], x] (* G. C. Greubel, May 05 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(2/(7*sqrt(1-4*x) -5)) \\ G. C. Greubel, May 05 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(7*Sqrt(1-4*x) -5) )); // G. C. Greubel, May 05 2019
(Sage) (2/(7*sqrt(1-4*x) -5)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 05 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Feb 14 2007
EXTENSIONS
a(16) corrected by G. C. Greubel, May 05 2019
STATUS
approved