A014228 -id:A014228 - cp's OEIS frontend

A259711 Numbers that can be written as a product of Catalan numbers.

1, 2, 4, 5, 8, 10, 14, 16, 20, 25, 28, 32, 40, 42, 50, 56, 64, 70, 80, 84, 100, 112, 125, 128, 132, 140, 160, 168, 196, 200, 210, 224, 250, 256, 264, 280, 320, 336, 350, 392, 400, 420, 429, 448, 500, 512, 528, 560, 588, 625, 640, 660, 672, 700, 784, 800, 840

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

nonn

			70 = 5*14 = C(3)*C(4). - _David A. Corneth_, Mar 26 2021

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Catalan Number

Cf. A000108, A001013, A003046, A005568, A014228, A259711, A342288.

A342288 a(n) = C(n)*C(n+2), where C(n) is the n-th Catalan number A000108(n).

Original entry on oeis.org

2, 5, 28, 210, 1848, 18018, 188760, 2085798, 24018280, 285817532, 3493769552, 43672119400, 556315613280, 7202300350500, 94561966954800, 1256886144107550, 16888191533829000, 229116465142280100, 3135277944052254000, 43238333101156993800, 600502564063273528800

Offset: 0

N. J. A. Sloane, Mar 26 2021

nonn

Inspired by A000108, A005568, A014228.

a[n_] := CatalanNumber[n] * CatalanNumber[n + 2]; Array[a, 21, 0] (* Amiram Eldar, Apr 02 2022 *)

A342288_list = [2]
for n in range(1,100): A342288_list.append(A342288_list[-1]*4*(2*n-1)*(2*n+3)//((n+1)*(n+3))) # Chai Wah Wu, Mar 26 2021

a(n) = a(n-1)*4*(2*n-1)*(2*n+3)/((n+1)*(n+3)). - Chai Wah Wu, Mar 26 2021

Sum_{n>=0} a(n)/4^(2*n+2) = 1/2 - 16/(15*Pi). - Amiram Eldar, Apr 02 2022

A014231 (Product of 3 successive Catalan numbers)/2.

Original entry on oeis.org

1, 5, 70, 1470, 38808, 1189188, 40489020, 1491345570, 58388438680, 2400295633736, 102692368441936, 4542162450316400, 206643434552856000, 9631060074695610000, 458381806260954003000, 22220282755463598704250, 1094733022441519300455000, 54717445279576990999935000

Offset: 0

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..560

Cf. A000108.

a:= proc(n) option remember; `if`(n=0, 1,
      8*(2*n-1)*(2*n+1)*(2*n+3)*a(n-1)/((n+1)*(n+2)*(n+3)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 20 2013

Times@@@Partition[CatalanNumber[Range[0,20]],3,1]/2 (* Harvey P. Dale, Apr 10 2013 *)

a(n) = A014228(n)/2.

cp's OEIS Frontend

A259711 Numbers that can be written as a product of Catalan numbers.

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A342288 a(n) = C(n)*C(n+2), where C(n) is the n-th Catalan number A000108(n).

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A014231 (Product of 3 successive Catalan numbers)/2.

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