A125230 Triangle T(n,k) (0<=k<=n) read by rows in which column k contains the binomial transform of the sequence of k 0's, (k+1) 1's, followed by 0's.

1, 1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 10, 11, 5, 1, 1, 15, 25, 16, 6, 1, 1, 21, 50, 42, 22, 7, 1, 1, 28, 91, 98, 64, 29, 8, 1, 1, 36, 154, 210, 163, 93, 37, 9, 1, 1, 45, 246, 420, 381, 256, 130, 46, 10, 1, 1, 55, 375, 792, 837, 638, 386, 176, 56, 11, 1, 1, 66, 550, 1419, 1749, 1485, 1024

Offset: 0

Author

Gary W. Adamson, Nov 24 2006

Keywords

• nonn
• tabl

Comments

A125231 is another triangle with the same row sums A045623: (1, 2, 5, 12, 28, 64, 144, 320...).

Examples

			T(5,2) = C(5,2) + C(5,3) + C(5,4) = 10 + 10 + 5 = 25.
First few rows of the triangle are:
1
1 1
1 3 1
1 6 4 1
1 10 11 5 1
1 15 25 16 6 1
		

Crossrefs

Cf. A007318, A125231. Columns k=0-3 give: A000012, A000217, A006522(n+1), A055796(n-3). Row sums give: A045623.

Programs

Maple

T:= (n, k)-> add (binomial (n, j), j=k..min(2*k, n)): seq (seq (T(n, k), k=0..n), n=0..12);

Formula

T(n,k) = Sum_{j=k..min(2*k,n)} C(n,j).

Extensions

Edited with more terms and Maple program by Alois P. Heinz, Oct 16 2009