|
%I #10 Aug 29 2019 17:13:42
%S 1,3,1,6,3,1,6,15,3,1,0,24,15,3,1,0,54,51,15,3,1,0,36,108,51,15,3,1,0,
%T 36,198,189,51,15,3,1,0,0,360,360,189,51,15,3,1,0,0,324,846,603,189,
%U 51,15,3,1,0,0,216,1296,1332,603,189,51,15,3,1,0,0,216,2484,2754,2061,603,189
%N Lower triangular array, called S1hat(-3), related to partition number array A145366.
%C If in the partition array M31hat(-3):=A145366 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-3). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
%C The first column is [1,3,6,6,0,0,0,...]= A008279(3,n-1), n>=1.
%H W. Lang, <a href="/A145367/a145367.txt">First 10 rows of the array and more.</a>
%H W. 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(-3;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(-3,n,1)= A008279(3,n-1) = [1,3,6,6,0,0,0,...], n>=1.
%e [1];[3,1];[6,3,1];[6,15,3,1];[0,24,15,3,1];...
%Y A145368 (row sums).
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_ Oct 17 2008
|
