OFFSET
1,2
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1724
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
FORMULA
a(n) = 2*A025235(n-2) for n>=2.
G.f.: (1+x-sqrt(1-2*x-7*x^2))/2. - Michael Somos, Jun 08 2000
G.f.: x + 2*x^2/G(0) with G(k) = (1 - x - 2*x^2/G(k+1)) (continued fraction). - Nikolaos Pantelidis, Dec 16 2022
From Peter Bala, May 01 2024: (Start)
O.g.f.: A(x) = x*S(x/(1 + 2*x)) = 2*x - x*S(- x/(1 - 4*x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318.
The g.f. satisfies A(x)^2 - (1 + x)*A(x) + x*(1 + 2*x) = 0.
A(x) = x*(1 + 2*x)/(1 + x - x*(1 + 2*x)/(1 + x - x*(1 + 2*x)/(1 + x - ...))).
A(x) = x/(1 - 2*x/(1 + 2*x - x/(1 - 2*x/(1 + 2*x - x/(1 - 2*x/(1 + 2*x - x/(1 - ...))))))). (End)
MATHEMATICA
Rest@ CoefficientList[Series[(1 + x - Sqrt[1 - 2 x - 7 x^2])/2, {x, 0, 27}], x] (* Michael De Vlieger, Jan 02 2020 *)
PROG
(PARI) a(n)=polcoeff((1+x-sqrt(1-2*x-7*x^2+x*O(x^n)))/2, n)
(Magma) a:=[1, 2]; for n in [3..30] do Append(~a, &+[a[k]*a[n-k]:k in [1..n-2]] ); end for; a; // Marius A. Burtea, Jan 02 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Corrected by T. D. Noe, Oct 31 2006
STATUS
approved