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A187357 Catalan trisection: A000108(3*n), n>=0. 13
1, 5, 132, 4862, 208012, 9694845, 477638700, 24466267020, 1289904147324, 69533550916004, 3814986502092304, 212336130412243110, 11959798385860453492, 680425371729975800390, 39044429911904443959240, 2257117854077248073253720, 131327898242169365477991900, 7684785670514316385230816156, 451959718027953471447609509424 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..18.

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

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A171375 A191409 A003372 * A135758 A152794 A053517

Adjacent sequences:  A187354 A187355 A187356 * A187358 A187359 A187360

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Mar 09 2011

STATUS

approved

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Last modified August 17 16:12 EDT 2022. Contains 356189 sequences. (Running on oeis4.)