A104711 Triangle T(n,m) = sum_{k=m..n} A001263(k,m).

1, 2, 1, 3, 4, 1, 4, 10, 7, 1, 5, 20, 27, 11, 1, 6, 35, 77, 61, 16, 1, 7, 56, 182, 236, 121, 22, 1, 8, 84, 378, 726, 611, 218, 29, 1, 9, 120, 714, 1902, 2375, 1394, 365, 37, 1, 10, 165, 1254, 4422, 7667, 6686, 2885, 577, 46, 1, 11, 220, 2079, 9372, 21527, 26090, 16745

Offset: 1

Author

Gary W. Adamson, Mar 19 2005

Keywords

• nonn
• tabl

Comments

This summation over columns of the Narayana triangle could also be defined as a multiplication

of the Narayana triangle from the left by the lower-left triangle represented by the all-1 sequence A000012.

Examples

			First few rows of the triangle are:
1;
2, 1;
3, 4, 1;
4, 10, 7, 1;
5, 20, 27, 11, 1;
6, 35, 77, 61, 16, 1;
...
		

Crossrefs

Cf. A014138, A104710, A005585, A000124.

Programs

Python

from sympy import binomial
def A001263(n,m):
    return binomial(n-1,m-1)*binomial(n,m-1)//m
def A104711(n,m):
    a = 0
    for k in range(m,n+1):
        a += A001263(k,m)
    return a
print([A104711(n,m) for n in range(20) for m in range(1,n+1)]) # R. J. Mathar, Oct 11 2009

Formula

Row sums: sum_{m=1..n} T(n,m) = A014138(n).

Extensions

Extended by R. J. Mathar, Oct 11 2009