A214403 Triangle, read by rows of terms T(n,k) for k=0..n^2, that starts with a '1' in row 0 with row n>0 consisting of 2*n-1 '1's followed by the partial sums of the prior row.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 36, 47, 62, 83, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 19, 24, 30, 37, 45, 55, 68, 85, 107, 135, 171, 218, 280, 363

Offset: 0

Author

Paul D. Hanna, Jul 15 2012

Keywords

• nonn
• tabf

Comments

Right border and row sums form A178325.

Examples

			Triangle begins:
  [1];
  [1, 1];
  [1,1,1, 1, 2];
  [1,1,1,1,1, 1,2,3, 4, 6];
  [1,1,1,1,1,1,1, 1,2,3,4,5, 6,8,11, 15, 21];
  [1,1,1,1,1,1,1,1,1, 1,2,3,4,5,6,7, 8,10,13,17,22, 28,36,47, 62, 83];
  ...
Row sums equal the row sums (A178325) of triangle A214398,
where A214398(n, k) = binomial(k^2+n-k-1, n-k):
  1;
  1, 1;
  1, 4, 1;
  1, 10, 9, 1;
  1, 20, 45, 16, 1;
  1, 35, 165, 136, 25, 1;
  1, 56, 495, 816, 325, 36, 1;
  1, 84, 1287, 3876, 2925, 666, 49, 1;
  ...
		

Links

• Paul D. Hanna, Rows n = 0..14, flattened.

Crossrefs

Cf. A131338, A178325, A214398.

Programs

PARI

{T(n, k)=if(k>n^2||n<0||k<0, 0, if(n==0,1,if(k<=2*n-1, 1, sum(i=0, k-2*n+1, T(n-1, i)))))}
for(n=0,10,for(k=0,n^2,print1(T(n,k),", "));print(""))