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.

Sequence in context: A122773 A029268 A064191 this_sequence A129033 A054090 A122517

Adjacent sequences: A127417 A127418 A127419 this_sequence A127421 A127422 A127423

nonn,tabl

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