A189175 Riordan matrix (1+x/sqrt(1-4*x),(1-sqrt(1-4*x))/2).
1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 20, 15, 9, 4, 1, 70, 49, 29, 14, 5, 1, 252, 168, 98, 49, 20, 6, 1, 924, 594, 342, 174, 76, 27, 7, 1, 3432, 2145, 1221, 627, 285, 111, 35, 8, 1, 12870, 7865, 4433, 2288, 1067, 440, 155, 44, 9, 1, 48620, 29172, 16302, 8437, 4004, 1716, 649, 209, 54, 10, 1
Offset: 0
Examples
Triangle begins: 1 1,1 2,2,1 6,5,3,1 20,15,9,4,1 70,49,29,14,5,1 252,168,98,49,20,6,1 924,594,342,174,76,27,7,1 3432,2145,1221,627,285,111,35,8,1
Programs
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Mathematica
T[n_,k_]=If[n==k,1,Binomial[2n-k,n-k](n^2+n k-k^2-k)/((2n-k)(2n-k-1))] Flatten[Table[T[n,k],{n,0,20},{k,0,n}]] -
Maxima
T(n,k):=if n=k then 1 else binomial(2*n-k,n-k)*(n^2+n*k-k^2-k)/((2*n-k)*(2*n-k-1)); create_list(T(n,k),n,0,20,k,0,n);
Formula
T(n,k)=[x^n] (1+x/sqrt(1-4*x))*((1-sqrt(1-4*x))/2)^k.
T(n,k) = binomial(2*n-k,n-k)*(n^2+n*k-k^2-k)/((2*n-k)*(2*n-k-1)) for k<=n, (n,k) <> (0,0), (1,1).
Recurrence: T(n+1,k+1) = T(n,k) + T(n,k+1) + ... + T(n,n).
Comments