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

0,8

Comments

Right border and row sums form A178325.

Links

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

Example

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];
[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];
[1,1,1,1,1,1,1,1,1,1,1,1,1, 1,2,3,4,5,6,7,8,9,10,11, 12,14,17,21,26,32,39,47,56, 66,78,93,112,136,166,203, 248,303,371,456,563, 698,869,1087, 1367, 1730]; ...
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; ...

Prog

(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(""))

Crossrefs

Cf. A131338, A178325, A214398.

Sequence in context: A332013 A324350 A175466 * A261527 A292436 A184097

Adjacent sequences: A214400 A214401 A214402 * A214404 A214405 A214406

Keyword

nonn,tabf

Author

Paul D. Hanna, Jul 15 2012

Status

approved