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Triangle read by rows: Catalan number C(n) repeated n+1 times.
4

%I #30 Nov 04 2024 17:15:53

%S 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,

%T 132,132,132,429,429,429,429,429,429,429,429,1430,1430,1430,1430,1430,

%U 1430,1430,1430,1430,4862,4862,4862,4862,4862,4862,4862,4862,4862,4862

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

%C 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

%F T(n,k) = A000108(n). - _R. J. Mathar_, Nov 03 2016

%F 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

%F Sum_{n>=0} 1/a(n) = 4 + 28*Pi/(27*sqrt(3)). - _Amiram Eldar_, Aug 18 2022

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 2, 2;

%e 5, 5, 5, 5 ;

%e 14, 14, 14, 14, 14;

%e 42, 42, 42, 42, 42, 42;

%e From _Philippe Deléham_, May 22 2015: (Start)

%e A = square array A039599, completed with zeros.

%e 1.....0.....0.....0...

%e 1.....1.....0.....0...

%e 2.....3.....1.....0...

%e 5.....9.....5.....1...

%e ......................

%e B = transpose of A.

%e 1.....1.....2.....5...

%e 0.....1.....3.....9...

%e 0.....0.....1.....5...

%e 0.....0.....0.....1...

%e ......................

%e A x B = this sequence read as square array.

%e 1.....1.....2.....5...

%e 1.....2.....5....14...

%e 2.....5....14....42...

%e 5....14....42...132...

%e ...................... (End)

%t Table[PadRight[{}, n + 1, CatalanNumber[n]], {n, 0, 8}] // Flatten (* _Amiram Eldar_, Aug 18 2022, after _Harvey P. Dale_ at A172417 *)

%o (Python)

%o from math import isqrt

%o from sympy import catalan

%o def A168256(n): return catalan((isqrt(n+1<<3)+1>>1)-1) # _Chai Wah Wu_, Nov 04 2024

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

%K nonn,tabl,easy,changed

%O 0,4

%A _Mark Dols_, Nov 21 2009