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%I #19 Aug 29 2019 16:45:33
%S 1,5,1,20,15,1,60,80,75,30,1,120,300,1000,200,375,50,1,120,720,4500,
%T 4000,900,6000,1875,400,1125,75,1,0,840,12600,42000,2520,31500,28000,
%U 52500,2100,21000,13125,700,2625,105,1,0,0,16800,134400,126000,3360,100800,336000
%N Partition number array, called M31(-5), related to A049411(n,m) = S1(-5;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(-5;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=5) in the family M31(-K) of partition number arrays.
%C If M31(-5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-5) := A049411.
%H W. Lang, <a href="/A144879/a144879.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_{j=1..n} (e(n,k,j)!*j!^e(n,k,j))) * Product_{j=1..n} S1(-5;j,1)^e(n,k,j) = M3(n,k) * Product_{j=1..n} S1(-5;j,1)^e(n,k,j), with S1(-5;n,1) = A008279(5,n-1)= [1,5,20,60,120,120,0,...], 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]; [5,1]; [20,15,1]; [60,80,75,30,1]; [120,300,1000,200,375,50,1]; ...
%e a(4,3) = 75 = 3*S1(-5;2,1)^2. The relevant partition of 4 is (2^2).
%Y Cf. A049428 (row sums).
%Y Cf. A144878 (M31(-4) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008
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