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%I #13 Aug 29 2019 15:55:06
%S 1,3,1,6,9,1,6,24,27,18,1,0,30,180,60,135,30,1,0,0,270,360,90,1080,
%T 405,120,405,45,1,0,0,0,1260,0,1890,2520,5670,210,3780,2835,210,945,
%U 63,1,0,0,0,0,1260,0,0,10080,11340,30240,0,7560,10080,45360,8505,420,10080,11340
%N Partition number array, called M31(-3), related to A049410(n,m) = S1(-3;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(-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 First member (K=3) in the family M31(-K) of partition number arrays.
%C If M31(-3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-3) := A049410.
%H W. Lang, <a href="/A144877/a144877.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(-3;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-3;j,1)^e(n,k,j),j=1..n) with S1(-3;n,1)|= A008279(3,n-1)= [1,3,6,6,0,...], 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]; [3,1]; [6,9,1]; [6,24,27,18,1]; [0,30,180,60,135,30,1]; ...
%e a(4,3) = 27 = 3*S1(-3;2,1)^2. The relevant partition of 4 is (2^2).
%Y Cf. A049426 (row sums).
%Y Cf. A144358 (M31(-2) array), A144878 (M31(-4) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008
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