A168256 Triangle read by rows: Catalan number C(n) repeated n+1 times.
1, 1, 1, 2, 2, 2, 5, 5, 5, 5, 14, 14, 14, 14, 14, 42, 42, 42, 42, 42, 42, 132, 132, 132, 132, 132, 132, 132, 429, 429, 429, 429, 429, 429, 429, 429, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 1430, 4862, 4862, 4862, 4862, 4862, 4862, 4862, 4862, 4862, 4862
Offset: 0
Examples
Triangle begins: 1; 1, 1; 2, 2, 2; 5, 5, 5, 5 ; 14, 14, 14, 14, 14; 42, 42, 42, 42, 42, 42; From _Philippe Deléham_, May 22 2015: (Start) A = square array A039599, completed with zeros. 1.....0.....0.....0... 1.....1.....0.....0... 2.....3.....1.....0... 5.....9.....5.....1... ...................... B = transpose of A. 1.....1.....2.....5... 0.....1.....3.....9... 0.....0.....1.....5... 0.....0.....0.....1... ...................... A x B = this sequence read as square array. 1.....1.....2.....5... 1.....2.....5....14... 2.....5....14....42... 5....14....42...132... ...................... (End)
Programs
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Mathematica
Table[PadRight[{}, n + 1, CatalanNumber[n]], {n, 0, 8}] // Flatten (* Amiram Eldar, Aug 18 2022, after Harvey P. Dale at A172417 *) -
Python
from math import isqrt from sympy import catalan def A168256(n): return catalan((isqrt(n+1<<3)+1>>1)-1) # Chai Wah Wu, Nov 04 2024
Formula
T(n,k) = A000108(n). - R. J. Mathar, Nov 03 2016
G.f.: (x*C(x)-x*y*C(x*y))/(x-x*y), where C(x) is the g.f. of A000108. - Vladimir Kruchinin, Nov 19 2020
Sum_{n>=0} 1/a(n) = 4 + 28*Pi/(27*sqrt(3)). - Amiram Eldar, Aug 18 2022
Comments