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%I #13 Aug 29 2019 15:56:04
%S 1,4,1,12,12,1,24,48,48,24,1,24,120,480,120,240,40,1,0,144,1440,1440,
%T 360,2880,960,240,720,60,1,0,0,2016,10080,504,10080,10080,20160,840,
%U 10080,6720,420,1680,84,1,0,0,0,16128,20160,0,16128,80640,80640,161280,1344,40320
%N Partition number array, called M31(-4), related to A049424(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 First 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) := A049424.
%H W. Lang, <a href="/A144878/a144878.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)= A008279(4,n-1)= [1,4,12,12,24,24,0,...], 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]; [4,1]; [12,12,1]; [24,48,48,24,1]; [24,120,480,120,240,40,1]; ...
%e a(4,3) = 48 = 3*S1(-4;2,1)^2. The relevant partition of 4 is (2^2).
%Y Cf. A049427 (row sums).
%Y Cf. A144877 (M31(-3) array), A144879 (M31(-5) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008