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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 15 14:07 EDT 2018. Contains 316236 sequences. (Running on oeis4.)