%I #33 Nov 07 2024 20:39:43

%S 1,2,2,5,5,5,14,14,14,14,42,42,42,42,42,132,132,132,132,132,132,429,

%T 429,429,429,429,429,429,1430,1430,1430,1430,1430,1430,1430,1430,4862,

%U 4862,4862,4862,4862,4862,4862,4862,4862,16796,16796,16796,16796,16796

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

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

%H Marc Chamberland, <a href="https://doi.org/10.1016/j.laa.2011.08.030">Factored matrices can generate combinatorial identities</a>, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

%F T(n,k) = A000108(n). - _R. J. Mathar_, Nov 03 2016

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

%e Triangle begins:

%e .....1

%e ....2,2

%e ...5,5,5

%e 14,14,14,14

%t Table[PadRight[{},n,CatalanNumber[n]],{n,10}]//Flatten (* _Harvey P. Dale_, Jun 05 2021 *)

%o (Python)

%o from math import isqrt

%o from sympy import catalan

%o def A172417(n): return catalan((m:=isqrt(k:=n<<1))+(k>m*(m+1))) # _Chai Wah Wu_, Nov 07 2024

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

%K nonn,tabl,easy,less

%O 1,2

%A _Mark Dols_, Feb 02 2010

%E Definition corrected by _R. J. Mathar_, Nov 03 2016