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%I #9 Aug 30 2019 04:03:50
%S 1,1,1,2,1,1,6,3,1,1,24,8,3,1,1,120,34,9,3,1,1,720,156,36,9,3,1,1,
%T 5040,924,166,37,9,3,1,1,40320,6144,968,168,37,9,3,1,1,362880,48096,
%U 6372,978,169,37,9,3,1,1,3628800,420480,49368,6416,980,169,37,9,3,1,1,39916800,4134240
%N Lower triangular array called S1hat(1) related to partition number array A107106.
%C If in the partition array M31hat(1):=A107106 entries 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 three columns are A000142(n-1) (factorials), A024419 (guess), A144352.
%H W. Lang, <a href="/A144351/a144351.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(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)|= |A008275(n,1)| = A000142(n-1) = (n-1)!.
%e [1];[1,1];[2,1,1];[6,3,1,1];[24,8,3,1,1];...
%Y Row sums A107107.
%Y A134134 (S1hat(2)= S2'(2)).
%K nonn,easy,tabl
%O 1,4
%A _Wolfdieter Lang_ Oct 09 2008
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