A168256 -id:A168256 - cp's OEIS frontend

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

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

A172414 Triangle read by rows: Catalan number C(n) repeated 2*n+1 times.

Original entry on oeis.org

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

Offset: 0

Mark Dols, Feb 02 2010

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

Cf. A000108, A001700 (row sums), A172417, A168256.

Table[PadRight[{}, 2*n + 1, CatalanNumber[n]], {n, 0, 8}] // Flatten (* Amiram Eldar, Aug 18 2022, after Harvey P. Dale at A172417 *)

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

Sum_{n>=0} 1/a(n) = 6 + 44*Pi/(27*sqrt(3)). - Amiram Eldar, Aug 18 2022

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

Data corrected and extended by Amiram Eldar, Aug 18 2022

A339028 Triangle T(n,m) = 3(m+1)C(n+m+3,m)C(2n+2,n-m)/((n+m+3)*C(n+1,m)).

Original entry on oeis.org

1, 3, 3, 9, 12, 9, 28, 42, 42, 28, 90, 144, 162, 144, 90, 297, 495, 594, 594, 495, 297, 1001, 1716, 2145, 2288, 2145, 1716, 1001, 3432, 6006, 7722, 8580, 8580, 7722, 6006, 3432, 11934, 21216, 27846, 31824, 33150, 31824, 27846, 21216, 11934, 41990, 75582, 100776, 117572, 125970, 125970, 117572, 100776, 75582, 41990

Offset: 0

Vladimir Kruchinin, Nov 20 2020

			1,
3,3,
9,12,9,
28,42,42,28,
90,144,162,144,
90,297,495,594,594,495,297

Cf. A000108, A000245, A002057 (row sums), A168256, A120406,

Maxima
```
T(n,m):=if n
				
```

G.f. ((sqrt(1-4*x*y)-sqrt(1-4*x))/(2*(x-x*y)))^3.

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

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A172414 Triangle read by rows: Catalan number C(n) repeated 2*n+1 times.

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A339028 Triangle T(n,m) = 3(m+1)C(n+m+3,m)C(2n+2,n-m)/((n+m+3)*C(n+1,m)).

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cp's OEIS Frontend

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

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A172414 Triangle read by rows: Catalan number C(n) repeated 2*n+1 times.

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Mathematica

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A339028 Triangle T(n,m) = 3*(m+1)*C(n+m+3,m)*C(2*n+2,n-m)/((n+m+3)*C(n+1,m)).

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Formula

A339028 Triangle T(n,m) = 3(m+1)C(n+m+3,m)C(2n+2,n-m)/((n+m+3)*C(n+1,m)).