A187359 Catalan trisection: A000108(3*n + 2)/2, n>=0.

1, 21, 715, 29393, 1337220, 64822395, 3282060210, 171529806825, 9183676536076, 501121108325684, 27767032438524099, 1558142747453650631, 88366931393503350700, 5056959295818949067010, 291650059796498346544020, 16934386878595523443214745, 989130828878080326811887228, 58078935727891217125276922940, 3426228463922436748774829232156, 202972497563788492865321721683556

Offset: 0

Author

Wolfdieter Lang, Mar 09 2011

Keywords

• nonn
• easy

Comments

See the comment under A187357 for the o.g.f.s of the general trisection of a sequence.

The sequence C(3*n+2) starts as 2, 42, 1430, 58786, 2674440, 129644790, 6564120420, 343059613650, ...

Crossrefs

Cf. A000108, A024492, A048990, A187357 (C(3*n)), A187358 (C(3*n+1)).

Programs

Mathematica

Table[CatalanNumber[3*n+2]/2, {n, 0, 20}] (* Amiram Eldar, Mar 16 2022 *)

Formula

a(n) = C(3*n+2)/2, n>=0, with C(n) = A000108(n).

O.g.f.: (3 - sqrt(1 - 4*x^(1/3)) - sqrt(2)*sqrt(sqrt(1 + 4*x^(1/3) + 16*x^(2/3)) +

(1 + 2*x^(1/3))))/(12*x).

From Ilya Gutkovskiy, Jan 21 2017: (Start)

E.g.f.: 3F3(5/6,7/6,3/2; 4/3,5/3,2; 64*x).

a(n) ~ 8^(2*n+1)/(3*sqrt(3*Pi)*n^(3/2)). (End)

Sum_{n>=0} a(n)/4^n = 1 - sqrt(3+2*sqrt(3))/3. - Amiram Eldar, Mar 16 2022

a(n) = (1/2)*Product_{1 <= i <= j <= 3*n+1} (3*i + j + 2)/(3*i + j - 1). - Peter Bala, Feb 22 2023