proposed

approved

editing

proposed

nonn,tabltabf

approved

editing

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jan 14 2007

nonn,tabl,new

Paul D . Hanna (pauldhanna(AT)juno.com), Jan 14 2007

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

0,4

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).

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.

Cf. A091352, A091351, A125781, A127419.

nonn,tabl

Paul D Hanna (pauldhanna(AT)juno.com), Jan 14 2007

approved