A104730 Triangle read by rows: T(n,k)=C(n+1,k)-C(k,n-k+1).

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 5, 10, 7, 1, 1, 6, 15, 19, 9, 1, 1, 7, 21, 35, 31, 11, 1, 1, 8, 28, 56, 69, 46, 13, 1, 1, 9, 36, 84, 126, 121, 64, 15, 1, 1, 10, 45, 120, 210, 251, 195, 85, 17, 1, 1, 11, 55, 165, 330, 462, 456, 295, 109, 19, 1, 1, 12, 66

Offset: 1

Author

Gary W. Adamson, Mar 20 2005

Keywords

• nonn
• tabl

Comments

Row sums are A027934: 1, 2, 5, 11, 24, 51, 107... Diagonal sums are A131298.

Examples

			The first few rows of the triangle are:
1;
1, 1;
1, 3, 1;
1, 4, 5, 1;
1, 5, 10, 7, 1;
1, 6, 15, 19, 9, 1;
1, 7, 31, 35, 31, 11, 1;
...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A027934, A026729, A007318.

Programs

Mathematica

Table[Binomial[n+1,k]-Binomial[k,n-k+1],{n,0,20},{k,0,n}]//Flatten (* Harvey P. Dale, Jan 16 2024 *)

Formula

Perform the operation A - B; then extract the triangle after deleting all zeros. P = infinite lower triangular Pascal's triangle matrix (A007318); B = A026729, as an infinite lower triangular matrix: [1; 0, 1;, 0, 1, 1; 0, 0, 2, 1; 0, 0, 1, 3, 1;...].

Extensions

Better definition from Paul Barry, Jun 26 2007

More terms from Harvey P. Dale, Jan 16 2024