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%I #10 Aug 29 2019 17:10:55
%S 1,3,1,12,3,1,60,12,9,3,1,360,60,36,12,9,3,1,2520,360,180,144,60,36,
%T 27,12,9,3,1,20160,2520,1080,720,360,180,144,108,60,36,27,12,9,3,1,
%U 181440,20160,7560,4320,3600,2520,1080,720,540,432,360,180,144,108,81,60,36,27
%N Partition number array, called M31hat(3).
%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) =: M31hat(3;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 This is the third (K=3) member of a family of partition number arrays: A107106, A134133,...
%H W. Lang, <a href="/A144880/a144880.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)= product(|S1(3;j,1)|^e(n,k,j),j=1..n) with |S1(3;n,1)|= A046089(1,n) = [1,3,12,60,...], 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.
%e [1];[3,1];[12,3,1];[60,12,9,3,1];[360,60,36,12,9,3,1];...
%e a(4,3)= 9 = |S1(3;2,1)|^2. The relevant partition of 4 is (2^2).
%Y A144882 (row sums).
%Y A134133 (M31hat(2) array). A144885 (M31hat(4) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008