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%I #9 Aug 29 2019 17:11:40
%S 1,3,1,12,3,1,60,21,3,1,360,96,21,3,1,2520,684,123,21,3,1,20160,4320,
%T 792,123,21,3,1,181440,35640,5292,873,123,21,3,1,1814400,293760,42768,
%U 5616,873,123,21,3,1,19958400,2881440,348840,45684,5859,873,123,21,3,1,239500800
%N Lower triangular array called S1hat(3) related to partition number array A144880.
%C If in the partition array M31hat(3):=A144880 entries 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 columns are A001710(n+1), A144883, A144884,...
%H W. Lang, <a href="/A144881/a144881.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)|= A046089(n,1) = A001710(n+1) = (n+1)!/2.
%e [1];[3,1];[12,3,1];[60,21,3,1];[360,96,21,3,1];...
%Y A144882 (row sums).
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008