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

%I A168256 #30 Nov 04 2024 17:15:53
%S A168256 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 A168256 132,132,132,429,429,429,429,429,429,429,429,1430,1430,1430,1430,1430,
%U A168256 1430,1430,1430,1430,4862,4862,4862,4862,4862,4862,4862,4862,4862,4862
%N A168256 Triangle read by rows: Catalan number C(n) repeated n+1 times.
%C A168256 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 A168256 T(n,k) = A000108(n). - _R. J. Mathar_, Nov 03 2016
%F A168256 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 A168256 Sum_{n>=0} 1/a(n) = 4 + 28*Pi/(27*sqrt(3)). - _Amiram Eldar_, Aug 18 2022
%e A168256 Triangle begins:
%e A168256   1;
%e A168256   1, 1;
%e A168256   2, 2, 2;
%e A168256   5, 5, 5, 5 ;
%e A168256   14, 14, 14, 14, 14;
%e A168256   42, 42, 42, 42, 42, 42;
%e A168256 From _Philippe Deléham_, May 22 2015: (Start)
%e A168256 A = square array A039599, completed with zeros.
%e A168256   1.....0.....0.....0...
%e A168256   1.....1.....0.....0...
%e A168256   2.....3.....1.....0...
%e A168256   5.....9.....5.....1...
%e A168256   ......................
%e A168256 B = transpose of A.
%e A168256   1.....1.....2.....5...
%e A168256   0.....1.....3.....9...
%e A168256   0.....0.....1.....5...
%e A168256   0.....0.....0.....1...
%e A168256   ......................
%e A168256 A x B = this sequence read as square array.
%e A168256   1.....1.....2.....5...
%e A168256   1.....2.....5....14...
%e A168256   2.....5....14....42...
%e A168256   5....14....42...132...
%e A168256   ...................... (End)
%t A168256 Table[PadRight[{}, n + 1, CatalanNumber[n]], {n, 0, 8}] // Flatten (* _Amiram Eldar_, Aug 18 2022, after _Harvey P. Dale_ at A172417 *)
%o A168256 (Python)
%o A168256 from math import isqrt
%o A168256 from sympy import catalan
%o A168256 def A168256(n): return catalan((isqrt(n+1<<3)+1>>1)-1) # _Chai Wah Wu_, Nov 04 2024
%Y A168256 Cf. A000108, A000984 (row sums), A039599, A172414, A172417.
%K A168256 nonn,tabl,easy
%O A168256 0,4
%A A168256 _Mark Dols_, Nov 21 2009