OFFSET
0,2
COMMENTS
Trisection of a sequence, given by its real o.g.f. G(x), is accomplished by
G(x) = G0(x^3) + x*G1(x^3) + (x^2)*G2(x^3), with the following solutions (using r := exp(2*Pi*i/3) = (-1 + sqrt(3)*i)/2):
G0(x) = (G(x^(1/3) + (G(r*x^(1/3)) + c.c.))/3,
G1(x) = (G(x^(1/3)) + ((1/r)*G(r*x^(1/3)) + c.c.))/(3*x^(1/3)),
G2(x) = (G(x^(1/3)) + (r*G(r*x^(1/3)) + c.c.))/(3*x^(2/3)),
where c.c. denotes the complex conjugate of the preceding expression.
See also the J. Arndt link, sect. 36.1.4,p.688: "Multisection by selecting terms with exponents s mod M", with M=3, where the o.g.f.s for the M-sected sequences with interspersed zeros are given for the general case.
LINKS
Joerg Arndt, Fxtbook.
FORMULA
a(n) = C(3*n), n >= 0, with C(n):= A000108(n) (Catalan).
O.g.f.: G0(x) = (sqrt(2*sqrt(1 + 4*x^(1/3) + 16*x^(2/3)) - (1 - 4*x^(1/3))) - sqrt(1 - 4*x^(1/3)))/(6*x^(1/3)).
From Ilya Gutkovskiy, Jan 13 2017: (Start)
E.g.f.: 3F3(1/6,1/2,5/6; 2/3,1,4/3; 64*x).
a(n) ~ 64^n/(3*sqrt(3*Pi)*n^(3/2)). (End)
D-finite with recurrence n*(3*n-1)*(3*n+1)*a(n) -8*(6*n-5)*(6*n-1)*(2*n-1)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
Sum_{n>=0} a(n)/4^n = (4/3)^(3/4) (A208745). - Amiram Eldar, Mar 16 2022
a(n) = Product_{1 <= i <= j <= 3*n-1} (3*i + j + 2)/(3*i + j - 1). - Peter Bala, Feb 22 2023
MATHEMATICA
Table[CatalanNumber[3*n], {n, 0, 20}] (* Amiram Eldar, Mar 16 2022 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Mar 09 2011
STATUS
approved