A187357 Catalan trisection: A000108(3*n), n >= 0.

1, 5, 132, 4862, 208012, 9694845, 477638700, 24466267020, 1289904147324, 69533550916004, 3814986502092304, 212336130412243110, 11959798385860453492, 680425371729975800390, 39044429911904443959240, 2257117854077248073253720, 131327898242169365477991900, 7684785670514316385230816156, 451959718027953471447609509424

Offset: 0

Author

Wolfdieter Lang, Mar 09 2011

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000108, A024492, A048990, A187358 (C(3*n+1)), A187359 (C(3*n+2)/2), A208745.

Programs

Mathematica

Table[CatalanNumber[3*n], {n, 0, 20}] (* Amiram Eldar, Mar 16 2022 *)

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