A131338 Triangle, read by rows of n*(n+1)/2 + 1 terms, that starts with a '1' in row 0 with row n consisting of n '1's followed by the partial sums of the prior row.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 3, 4, 6, 9, 14, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 29, 43, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 27, 37, 51, 71, 100, 143, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 35, 46, 61, 81, 108, 145, 196

Offset

0,7

Links

Paul D. Hanna, Rows n = 0..16, flattened.

Formula

T(n,k) = Sum_{i=0..k-n} T(n-1,i) for k>n, else T(n,k)=1 for n>=k>=0.

Right border: T(n+1, (n+1)*(n+2)/2) = A098569(n) = Sum_{k=0..n} C( (k+1)*(k+2)/2 + n-k-1, n-k).

T(n, n*(n-1)/2 + 1) = Sum_{k=0..n-1} C(k*(k+1)/2, n-k) = A121690(n-1) for n>=1. - Paul D. Hanna, Aug 30 2007

Example

Triangle begins:
1;
1, 1;
1,1, 1,2;
1,1,1, 1,2,3,5;
1,1,1,1, 1,2,3,4,6,9,14;
1,1,1,1,1, 1,2,3,4,5,7,10,14,20,29,43;
1,1,1,1,1,1, 1,2,3,4,5,6,8,11,15,20,27,37,51,71,100,143;
1,1,1,1,1,1,1, 1,2,3,4,5,6,7,9,12,16,21,27,35,46,61,81,108,145,196,267,367,510; ...
Row sums equal the row sums (A098569) of triangle A098568,
where A098568(n, k) = binomial( (k+1)*(k+2)/2 + n-k-1, n-k):
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 10, 21, 10, 1;
1, 15, 56, 55, 15, 1;
1, 21, 126, 220, 120, 21, 1; ...

Prog

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

Crossrefs

Cf. A098568, A098569 (row sums), A121690, A183202.

Cf. A214403 (variant).

Sequence in context: A293429 A367313 A201075 * A369995 A242784 A265313

Adjacent sequences: A131335 A131336 A131337 * A131339 A131340 A131341

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 29 2007

Status

approved