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%I #9 Aug 29 2019 15:57:47
%S 1,5,1,30,5,1,210,55,5,1,1680,360,55,5,1,15120,3630,485,55,5,1,151200,
%T 29820,4380,485,55,5,1,1663200,321300,39570,5005,485,55,5,1,19958400,
%U 3225600,421800,43320,5005,485,55,5,1,259459200,38808000,4265100,470550,46445,5005
%N Lower triangular array called S1hat(5) related to partition number array A144890.
%C If in the partition array M31hat(5):=A144890 entries with the same parts number m are summed one obtains this triangle of numbers S1hat(5). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
%C The first columns are A001720(n+3)=(n+3)!/4!, A144893, A144894,...
%H W. Lang, <a href="/A144891/a144891.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(5;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(5,n,1)|= A049353(n,1) = A001720(n+3) = (n+3)!/4!.
%e [1];[5,1];[30,5,1];[210,55,5,1];[1680,360,55,5,1];...
%Y A144892 (row sums).
%K nonn,easy,tabl
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008
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