|
|
A060642
|
|
Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.
|
|
22
|
|
|
1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 22, 21, 8, 1, 11, 43, 59, 36, 10, 1, 15, 80, 144, 124, 55, 12, 1, 22, 141, 321, 362, 225, 78, 14, 1, 30, 240, 669, 944, 765, 370, 105, 16, 1, 42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1, 56, 640, 2511, 5100, 6215, 4848, 2478, 824, 171, 20, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} A000041(n-k)*A(k;x)*x, A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004
G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j))^k. - Ilya Gutkovskiy, Feb 13 2021
|
|
EXAMPLE
|
Table begins:
1;
2, 1;
3, 4, 1;
5, 10, 6, 1;
7, 22, 21, 8, 1;
11, 43, 59, 36, 10, 1;
15, 80, 144, 124, 55, 12, 1;
22, 141, 321, 362, 225, 78, 14, 1;
30, 240, 669, 944, 765, 370, 105, 16, 1;
42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1;
...
For n=4 there are 5 partitions of 4, namely 4, 31, 22, 211, 11111. There are 5 ways to pick 1 of them; 10 ways to partition one of them into 2 ordered parts: 3,1; 1,3; 2,2; 21,1; 1,21; 2,11; 11,2; 111,1; 1,111; 11,11; 6 ways to partition one of them into 3 ordered parts: 2,1,1; 1,2,1; 1,1,2; 11,1,1; 1,11,1; 1,1,11; and one way to partition one of them into 4 ordered parts: 1,1,1,1. So row 4 is 5,10,6,1.
|
|
MAPLE
|
A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
A(n-j, k)*numtheory[sigma](j), j=1..n)/n)
end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
# Uses function PMatrix from A357368. Adds row and column for n, k = 0.
|
|
MATHEMATICA
|
A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[ Table[ T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|