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%I #14 Aug 29 2019 16:43:37
%S 1,4,1,20,12,1,120,80,48,24,1,840,600,800,200,240,40,1,6720,5040,7200,
%T 4000,1800,4800,960,400,720,60,1,60480,47040,70560,84000,17640,50400,
%U 28000,33600,4200,16800,6720,700,1680,84,1,604800,483840,752640,940800,504000,188160
%N Partition number array, called M31(4), related to A049352(n,m)= |S1(4;n,m)| (generalized Stirling triangle).
%C Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(4;n,k) with the k-th partition of n in A-St order.
%C The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C Fourth member (K=4) in the family M31(K) of partition number arrays.
%C If M31(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(4)|:= A049352.
%H W. Lang, <a href="/A144354/a144354.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,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(4;j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)|= A001715(n+2) = (n+2)!/3!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. M3(n,k)=A036040.
%e [1];[4,1];[20,12,1];[120,80,48,24,1];[840,600,800,200,240,40,1];...
%e a(4,3)= 48 = 3*|S1(4;2,1)|^2. The relevant partition of 4 is (2^2).
%Y A049377 (row sums).
%Y A144353 (M31(3) array), A144355 (M31(5) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008
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