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A124926 Triangle read by rows: T(n,k)=binom(n,k)*r(k), where r(k) are the Riordan numbers (r(k)=A005043(k); 0<=k<=n). 1

%I

%S 1,1,0,1,0,1,1,0,3,1,1,0,6,4,3,1,0,10,10,15,6,1,0,15,20,45,36,15,1,0,

%T 21,35,105,126,105,36,1,0,28,56,210,336,420,288,91,1,0,36,84,378,756,

%U 1260,1296,819,232,1,0,45,120,630,1512,3150,4320,4095,2320,603,1,0,55,165

%N Triangle read by rows: T(n,k)=binom(n,k)*r(k), where r(k) are the Riordan numbers (r(k)=A005043(k); 0<=k<=n).

%C Row sums = Catalan numbers, A000108: (1, 1, 2, 5, 14, 42...); e.g. sum of row 4 terms = A000108(4) = 14 = (1 + 0 + 6 + 4 + 3). A005043 is the inverse binomial transform of the Catalan numbers.

%D Chao-Jen Wang, Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords, http://people.brandeis.edu/~gessel/homepage/students/wangthesis.pdf.

%e First few rows of the triangle are:

%e 1;

%e 1, 0;

%e 1, 0, 1;

%e 1, 0, 3, 1;

%e 1, 0, 6, 4, 3;

%e 1, 0, 10, 10, 15, 6;

%e 1, 0, 15, 20, 45, 36, 15;

%e ...

%p r:=n->(1/(n+1))*sum((-1)^i*binomial(n+1,i)*binomial(2*n-2*i,n-i),i=0..n): T:=(n,k)->r(k)*binomial(n,k): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

%Y Cf. A005043, A000108.

%K nonn,tabl

%O 0,9

%A _Gary W. Adamson_, Nov 12 2006

%E Edited by _N. J. A. Sloane_, Nov 29 2006

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Last modified February 18 23:26 EST 2018. Contains 299330 sequences. (Running on oeis4.)