%I #15 Jun 23 2013 17:45:49

%S 1,3,1,9,4,1,29,13,5,1,99,42,18,6,1,351,141,60,24,7,1,1275,492,201,84,

%T 31,8,1,4707,1767,693,285,115,39,9,1,17577,6474,2460,978,400,154,48,

%U 10,1,66197,24051,8934,3438,1378,554,202,58,11,1,250953,90248,32985,12372,4816,1932,756,260,69,12,1

%N Riordan matrix (1/((1-x)*sqrt(1-4*x)),x/(1-x)).

%C Row sums are A082590.

%C First column is A006134.

%F a(n,k) = [x^n] 1/((1-x)*sqrt(1-4*x))*(x/(1-x))^k.

%F Recurrence: a(n+1,k+1) = a(n,k+1) + a(n,k).

%F a(n,k) = sum(binomial(n-i,k)*binomial(2*i,i),i=0..n).

%F G.f.: 1/(sqrt(1-4*x)*(1-x-x*y)).

%e Triangle begins:

%e 1,

%e 3,1,

%e 9,4,1,

%e 29,13,5,1,

%e 99,42,18,6,1,

%e 351,141,60,24,7,1,

%e 1275,492,201,84,31,8,1,

%t Select[Flatten[Table[Sum[Binomial[n-i,k]Binomial[2i,i],{i,0,n}],{n,0,10},{k,0,10}]],#!=0&] (* _Harvey P. Dale_, Jul 05 2012 *)

%o (Maxima) create_list(sum(binomial(n-i,k)*binomial(2*i,i), i,0,n),n,0,8,k,0,n);

%K nonn,easy,tabl

%O 0,2

%A _Emanuele Munarini_, Mar 15 2011

%E Mathematica program corrected by _Harvey P. Dale_, Jul 05 2012

%E Comment added and comment corrected by _Michel Marcus_, Jun 23 2013