%I #35 Oct 31 2021 07:46:15

%S 1,2,1,4,3,2,9,8,7,5,23,22,21,19,14,65,64,63,61,56,42,197,196,195,193,

%T 188,174,132,626,625,624,622,617,603,561,429,2056,2055,2054,2052,2047,

%U 2033,1991,1859,1430,6918,6917,6916,6914,6909,6895,6853,6721,6292

%N Triangle of partial sums of Catalan numbers.

%C Diagonal elements = Catalan numbers (A000108).

%C First column = partial sums of Catalan numbers (A014137).

%C Row sums = partial sums of central binomial coefficients (A006134).

%C Row square-sums = A182018. - _Emanuele Munarini_, Apr 06 2012

%C Central coefficients = A210670.

%H Vincenzo Librandi, <a href="/A210658/b210658.txt">Rows n = 0..100, flattened</a>

%F Recurrence: T(n+1,k+1) = T(n,k)+C(n+1)-C(k).

%F G.f. (C(x)-y*C(x*y))/((1-x)*(1-y)), where C(x)=(1-sqrt(1-4x))/(2x) is the generating series for the Catalan numbers.

%e Triangle begins:

%e 1,

%e 2, 1,

%e 4, 3, 2,

%e 9, 8, 7, 5,

%e 23, 22, 21, 19, 14,

%e 65, 64, 63, 61, 56, 42,

%e 197, 196, 195, 193, 188, 174, 132

%t Flatten[Table[Sum[Binomial[2i,i]/(i+1),{i,k,n}],{n,0,10},{k,0,n}]]

%o (Maxima) create_list(sum(binomial(2*i,i)/(i+1),i,k,n),n,0,10,k,0,n);

%Y Cf. A000108, A014137, A006134, A210670.

%K nonn,easy,tabl

%O 0,2

%A _Emanuele Munarini_, Mar 28 2012