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

%I A172417 #33 Nov 07 2024 20:39:43
%S A172417 1,2,2,5,5,5,14,14,14,14,42,42,42,42,42,132,132,132,132,132,132,429,
%T A172417 429,429,429,429,429,429,1430,1430,1430,1430,1430,1430,1430,1430,4862,
%U A172417 4862,4862,4862,4862,4862,4862,4862,4862,16796,16796,16796,16796,16796
%N A172417 Triangle read by rows: Catalan number C(n) repeated n times.
%C A172417 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 A172417 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 A172417 T(n,k) = A000108(n). - _R. J. Mathar_, Nov 03 2016
%F A172417 Sum_{n>=1} 1/a(n) = 2 + 16*Pi/(27*sqrt(3)). - _Amiram Eldar_, Aug 18 2022
%e A172417 Triangle begins:
%e A172417 .....1
%e A172417 ....2,2
%e A172417 ...5,5,5
%e A172417 14,14,14,14
%t A172417 Table[PadRight[{},n,CatalanNumber[n]],{n,10}]//Flatten (* _Harvey P. Dale_, Jun 05 2021 *)
%o A172417 (Python)
%o A172417 from math import isqrt
%o A172417 from sympy import catalan
%o A172417 def A172417(n): return catalan((m:=isqrt(k:=n<<1))+(k>m*(m+1))) # _Chai Wah Wu_, Nov 07 2024
%Y A172417 Cf. A001791 (row sums), A000108, A039598, A168256, A172414.
%K A172417 nonn,tabl,easy,less
%O A172417 1,2
%A A172417 _Mark Dols_, Feb 02 2010
%E A172417 Definition corrected by _R. J. Mathar_, Nov 03 2016