%I #16 Feb 22 2024 08:51:16

%S 1,1,1,0,1,1,0,1,1,1,0,0,1,1,1,0,0,1,1,1,1,0,0,0,1,1,1,1,0,0,0,1,1,1,

%T 1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,

%U 0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,1,1

%N Lower triangular array, called S1hat(-1), related to partition number array A145361.

%C 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.

%C The first column is [1,1,0,0,0,...]=A008279(1,n-1), n>=1.

%C a(n,m) gives the number of partitions of n with m parts, with each part not exceeding 2. - _Wolfdieter Lang_, Aug 03 2023

%H Wolfdieter Lang, <a href="/A145362/a145362.txt">First 10 rows of the array and more</a>.

%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Lang/lang.html">Combinatorial Interpretation of Generalized Stirling Numbers</a>, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

%F 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.

%F THe triangle starts in row n with ceiling(n/2) - 1 zeros, and is 1 otherwise. - _Wolfdieter Lang_, Aug 03 2023

%e Triangle begins:

%e [1];

%e [1,1];

%e [0,1,1];

%e [0,1,1,1];

%e [0,0,1,1,1];

%e [0,0,1,1,1,1];

%e ...

%Y Cf. A004526(n+2), n>=1, (row sums).

%Y Cf. A008275, A008279, A008284, A036039, A145361.

%K nonn,easy,tabl

%O 1,1

%A _Wolfdieter Lang_ Oct 17 2008