A143866 Eigentriangle of A027293.

1, 1, 1, 2, 1, 2, 3, 2, 2, 5, 5, 3, 4, 5, 12, 7, 5, 6, 10, 12, 29, 11, 7, 10, 15, 24, 29, 69, 15, 11, 14, 25, 36, 58, 69, 165, 22, 15, 22, 35, 60, 87, 138, 165, 393, 937, 42, 30, 44, 75, 132, 203, 345, 495, 786, 937, 2233, 56, 42, 60, 110, 180, 319, 483, 825, 1179, 1874, 2233, 5322

Offset: 1

Author

Gary W. Adamson, Sep 04 2008

Keywords

• nonn
• easy
• tabl

Comments

Left border = partition numbers, A000041 starting (1, 1, 2, 3, 5, 7, ...). Right border = INVERT transform of partition numbers starting (1, 1, 2, 5, 12, ...); with row sums the same sequence but starting (1, 2, 5, 12, ...). Sum of n-th row terms = rightmost term of next row.

For another definition of L-eigen-matrix of A027293 see A343234. - Wolfdieter Lang, Apr 16 2021

Examples

			The triangle begins:
n \ k    1  2  3  4   5   6   7   8   9  10   11 ...
-------------------------------------------
1:       1
2:       1  1
3:       2  1  2
4:       3  2  2  5
5:       5  3  4  5  12
6:       7  5  6 10  12  29
7:      11  7 10 15  24  29  69
8:      15 11 14 25  36  58  69 165
9:      22 15 22 35  60  87 138 165 393
10:     30 22 30 55  84 145 207 330 393 937
11:     42 30 44 75 132 203 345 495 786 937 2233
... reformatted and extended by _Wolfdieter Lang_, May 02 2021
Row 4 = (3, 2, 2, 5) = termwise product of (3, 2, 1, 1) and (1, 1, 2, 5) = (3*1, 2*1, 1*2, 1*5).
		

Crossrefs

Cf. A027293, A067687, A000041, A343234.

Formula

Triangle read by rows, A027293 * (A067687 * 0^(n-k)); 1 <= k <= n. (A067687 * 0^(n-k)) = an infinite lower triangular matrix with the INVERT transform of the partition function as the main diagonal: (1, 1, 2, 5, 12, 29, 69, 165, ...); and the rest zeros. Triangle A027293 = n terms of "partition numbers decrescendo"; by rows = termwise product of n terms of partition decrescendo and n terms of A027293: (1, 1, 2, 5, 12, 29, 69, 165, ...).