OFFSET
0,2
COMMENTS
See the comment under A187357 for the o.g.f.s for the general trisection of a sequence.
FORMULA
a(n) = C(3*n+1), n>=0, with C(n) = A000108(n) (Catalan).
O.g.f.: (sqrt(2*sqrt(1+4*x^(1/3)+16*x^(2/3))-(1+8*x^(1/3))) - sqrt(1-4*x^(1/3)))/(6*x^(2/3)).
From Ilya Gutkovskiy, Jan 13 2017: (Start)
E.g.f.: 3F3(1/2,5/6,7/6; 1,4/3,5/3; 64*x).
a(n) ~ 4^(3*n+1)/(3*sqrt(3*Pi)*n^(3/2)). (End)
Sum_{n>=0} a(n)/4^n = 2*sqrt(2*sqrt(3) - 3)/3. - Amiram Eldar, Mar 16 2022
a(n) = Product_{1 <= i <= j <= 3*n} (3*i + j + 2)/(3*i + j - 1). - Peter Bala, Feb 22 2023
MATHEMATICA
Table[CatalanNumber[3*n+1], {n, 0, 20}] (* Amiram Eldar, Mar 16 2022 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Mar 09 2011
STATUS
approved