OFFSET
1,1
COMMENTS
If in the partition array M31hat(-1):=A145361 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-1). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
The first column is [1,1,0,0,0,...]=A008279(1,n-1), n>=1.
a(n,m) gives the number of partitions of n with m parts, with each part not exceeding 2. - Wolfdieter Lang, Aug 03 2023
LINKS
Wolfdieter Lang, First 10 rows of the array and more.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
FORMULA
a(n,m) = sum(product(S1(-1;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S1(-1,n,1)= A008279(1,n-1) = [1,1,0,0,0,...], n>=1.
THe triangle starts in row n with ceiling(n/2) - 1 zeros, and is 1 otherwise. - Wolfdieter Lang, Aug 03 2023
EXAMPLE
Triangle begins:
[1];
[1,1];
[0,1,1];
[0,1,1,1];
[0,0,1,1,1];
[0,0,1,1,1,1];
...
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang Oct 17 2008
STATUS
approved