A172414 -id:A172414 - cp's OEIS frontend

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

Mark Dols, Nov 21 2009

As square array, it is A x B where A = square array A039599 (completed with zeros) and B = transpose of A. - Philippe Deléham, May 22 2015

			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)

Cf. A000108, A000984 (row sums), A039599, A172414, A172417.

Table[PadRight[{}, n + 1, CatalanNumber[n]], {n, 0, 8}] // Flatten (* Amiram Eldar, Aug 18 2022, after Harvey P. Dale at A172417 *)

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

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

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

Original entry on oeis.org

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

Mark Dols, Feb 02 2010

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

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

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

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

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

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

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

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

cp's OEIS Frontend

A168256 Triangle read by rows: Catalan number C(n) repeated n+1 times.

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