A128627 Triangle read by rows. Convolution triangle based on A002865.

1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 2, 3, 0, 1, 2, 5, 3, 4, 0, 1, 4, 6, 9, 4, 5, 0, 1, 4, 13, 12, 14, 5, 6, 0, 1, 7, 16, 28, 20, 20, 6, 7, 0, 1, 8, 30, 39, 50, 30, 27, 7, 8, 0, 1, 12, 40, 78, 76, 80, 42, 35, 8, 9, 0, 1, 14, 66, 115, 161, 130, 119, 56, 44, 9, 10, 0, 1

Offset: 1

Author

Alford Arnold, Mar 22 2007

Keywords

• nonn
• tabl

Comments

Triangular array illustrating the application of cyclic partitions to the computation of partitions of an integer into parts of k kinds (cf. A060850).

The array is constructed by summing sequences associated with each cyclic partition as indicated below: (n' here denotes the sum of preceding sequences).

4 1 2 3

22 1 3 6

4' 2 5 9

5 1 2 3 4

32 1 4 9 16

5' 2 6 12 20

6 1 2 3 4 5 6 7 8 9

42 1 4 9 16 25 36 49 64 81

33 1 3 6 10 15 21 28 36 45

222 1 4 10 20 35 56 84 120 165

6' 4 13 28 50 80 119 168 228 300

7 1 2 3 4 5 6 7 8 9

52 1 4 9 16 25 36 49 64 81

43 1 4 9 16 25 36 49 64 81

322 1 6 18 40 75 126 196 288 405

7' 4 16 39 76 130 204 301 424 576

8 1 2 3 4 5 6 7 8 9

62 1 4 9 16 25 36 49 64 81

53 1 4 9 16 25 36 49 64 81

44 1 3 6 10 15 21 28 36 45

422 1 6 18 40 75 126 196 288 405

332 1 6 18 40 75 126 196 288 405

2222 1 5 15 35 70 126 210 330 495

8' 7 30 78 161 290 477 735 1078 1521

Examples

			The diagonal 9th diagonal of A060850 is 22 185 810 2580 6765 ... and can be computed from a(n) and A007318 as illustrated:
   1
   0    1
   1    0    1
   1    2    0    1
   2    2    3    0
   2    5    3    4
   4    6    9    4
   4   13   12   14
   7   16   28   20
       30   39   50
            78   76
                161
times
   1
   1    9
   1    8   45
   1    7   36  165
   1    6   28  120
   1    5   21   84
   1    4   15   56
   1    3   10   35
   1    2    6   20
        1    3   10
             1    4
                  1
yields
   1
   0    9
   1    0   45
   1   14    0  165
   2   12   84    0
   2   25   63  336
   4   24  135  224
   4   39  120  490
   7   32  168  400
       30  117  500
            78  304
                161
summing to
  22  185  810 2580 ...
Triangle T(n, k) starts:
  [ 1] 1;
  [ 2] 0,  1;
  [ 3] 1,  0,  1;
  [ 4] 1,  2,  0,  1;
  [ 5] 2,  2,  3,  0,  1;
  [ 6] 2,  5,  3,  4,  0,  1;
  [ 7] 4,  6,  9,  4,  5,  0,  1;
  [ 8] 4, 13, 12, 14,  5,  6,  0,  1;
  [ 9] 7, 16, 28, 20, 20,  6,  7,  0,  1;
  [10] 8, 30, 39, 50, 30, 27,  7,  8,  0,  1;
		

Crossrefs

Cf. A002865, A060850, A007318.

Programs

Maple

# Using function A002865 and function PMatrix from A357368.
A128627Triangle := proc(dim) local M, Row, r;
M := PMatrix(dim, n -> A002865(n-1));
Row := r -> convert(linalg:-row(M, r), list)[2..r];
for r from 2 to dim do lprint(Row(r)) od end:
A128627Triangle(11); # Peter Luschny, Oct 03 2022

Extensions

New name by Peter Luschny, Oct 03 2022