A172417 Triangle read by rows: Catalan number C(n) repeated n times.

1, 2, 2, 5, 5, 5, 14, 14, 14, 14, 42, 42, 42, 42, 42, 132, 132, 132, 132, 132, 132, 429, 429, 429, 429, 429, 429, 429, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 4862, 4862, 4862, 4862, 4862, 4862, 4862, 4862, 4862, 16796, 16796, 16796, 16796, 16796

Offset: 1

Author

Mark Dols, Feb 02 2010

Keywords

• nonn
• tabl
• easy
• less

Comments

Read as a square array, we obtain the Hankel matrix ( 1/(i+j)*binomial(2*i+2*j-2, i+j-1) )A039598%20*%20transpose(A039598)%20(Cholesky%20factorization).%20See%20Chamberland,%20p.%201669.%20-%20_Peter%20Bala">i,j >= 1 equal to A039598 * transpose(A039598) (Cholesky factorization). See Chamberland, p. 1669. - _Peter Bala, Oct 15 2023

Examples

			Triangle begins:
.....1
....2,2
...5,5,5
14,14,14,14
		

Links

• Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

Crossrefs

Cf. A001791 (row sums), A000108, A039598, A168256, A172414.

Programs

Mathematica

Table[PadRight[{},n,CatalanNumber[n]],{n,10}]//Flatten (* Harvey P. Dale, Jun 05 2021 *)

Python

from math import isqrt
from sympy import catalan
def A172417(n): return catalan((m:=isqrt(k:=n<<1))+(k>m*(m+1))) # Chai Wah Wu, Nov 07 2024

Formula

T(n,k) = A000108(n). - R. J. Mathar, Nov 03 2016

Sum_{n>=1} 1/a(n) = 2 + 16*Pi/(27*sqrt(3)). - Amiram Eldar, Aug 18 2022

Extensions

Definition corrected by R. J. Mathar, Nov 03 2016