A127420 Triangle, read by rows, where row n+1 is generated from row n by first inserting zeros at positions {(m+2)*(m+3)/2, m>=0} in row n and then taking the partial sums in reverse order, for n>=2, starting with 1's in the initial two rows.

1, 1, 1, 2, 1, 1, 4, 2, 2, 1, 9, 5, 5, 3, 1, 1, 24, 15, 15, 10, 5, 5, 2, 1, 77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1, 295, 218, 218, 165, 112, 112, 74, 51, 28, 28, 15, 7, 4, 1, 1, 1329, 1034, 1034, 816, 598, 598, 433, 321, 209, 209, 135, 84, 56, 28, 28, 13, 6, 2, 1, 6934, 5605

Offset

0,4

Comments

Column 0 forms A091352, which also equals column 1 of table A125781, where table A125781 is generated by a complementary recurrence of this triangle. The number of terms in row n is A127419(n).

Links

Table of n, a(n) for n=0..70.

Example

To generate row 6, start with row 5:
24, 15, 15, 10, 5, 5, 2, 1;
insert zeros at positions [1,4,8,13,..., (m+2)*(m+3)/2 - 2,...]:
24, 0, 15, 15, 0, 10, 5, 5, 0, 2, 1;
then row 6 equals the partial sums of row 5 taken in reverse order:
24, _0, 15, 15, _0, 10, _5, 5, 0, 2, 1;
77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1.
Triangle begins:
1;
1, 1;
2, 1, 1;
4, 2, 2, 1;
9, 5, 5, 3, 1, 1;
24, 15, 15, 10, 5, 5, 2, 1;
77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1;
295, 218, 218, 165, 112, 112, 74, 51, 28, 28, 15, 7, 4, 1, 1;
1329, 1034, 1034, 816, 598, 598, 433, 321, 209, 209, 135, 84, 56, 28, 28, 13, 6, 2, 1;
Column 0 of this triangle equals column 1 of triangle A091351, where triangle A091351 begins:
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 9, 9, 4, 1;
1, 24, 30, 16, 5, 1;
1, 77, 115, 70, 25, 6, 1;
1, 295, 510, 344, 135, 36, 7, 1; ...
and column k of A091351 = row sums of matrix power A091351^k for k>=0.

Crossrefs

Cf. A091352, A091351, A125781, A127419.

Sequence in context: A176452 A244581 A064191 * A129033 A054090 A239456

Adjacent sequences: A127417 A127418 A127419 * A127421 A127422 A127423

Keyword

nonn,tabf

Author

Paul D. Hanna, Jan 14 2007

Status

approved