|
Search: id:A131338
|
|
|
| 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. |
|
+0
4
|
|
| 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
(list; table; graph; listen)
|
|
|
OFFSET
|
0,7
|
|
|
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 (pauldhanna(AT)juno.com), 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) = C( (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; ...
|
|
PROGRAM
|
(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))))}
|
|
CROSSREFS
|
Cf. A098568, A098569 (row sums).
Cf. A121690.
Adjacent sequences: A131335 A131336 A131337 this_sequence A131339 A131340 A131341
Sequence in context: A029384 A094102 A063746 this_sequence A106498 A093466 A125761
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Paul D. Hanna (pauldhanna(AT)juno.com), Jun 29 2007
|
|
|
Search completed in 0.008 seconds
|
