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A128627
Triangle read by rows. Convolution triangle based on A002865.
1
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
1,8
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
EXAMPLE
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;
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alford Arnold, Mar 22 2007
EXTENSIONS
New name by Peter Luschny, Oct 03 2022
STATUS
approved