A172417 - OEIS

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

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OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..50.

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

FORMULA

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

EXAMPLE

Triangle begins:

.....1

....2,2

...5,5,5

14,14,14,14

MATHEMATICA

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

PROG

(Python)

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

CROSSREFS

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

Sequence in context: A025510 A350172 A356387 * A158106 A185313 A208301

Adjacent sequences: A172414 A172415 A172416 * A172418 A172419 A172420

KEYWORD

nonn,tabl,easy,less

AUTHOR

Mark Dols, Feb 02 2010

EXTENSIONS

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

STATUS

approved