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

1,5

Comments

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

Links

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

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;...].

Example

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;
...

Mathematica

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

Crossrefs

Cf. A027934, A026729, A007318.

Sequence in context: A124020 A124234 A135226 * A249488 A275204 A321876

Adjacent sequences: A104727 A104728 A104729 * A104731 A104732 A104733

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 20 2005

Extensions

Better definition from Paul Barry, Jun 26 2007

More terms from Harvey P. Dale, Jan 16 2024

Status

approved