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%I #10 Aug 29 2019 17:53:49
%S 1,5,1,30,5,1,210,30,25,5,1,1680,210,150,30,25,5,1,15120,1680,1050,
%T 900,210,150,125,30,25,5,1,151200,15120,8400,6300,1680,1050,900,750,
%U 210,150,125,30,25,5,1,1663200,151200,75600,50400,44100,15120,8400,6300,5250,4500
%N Partition number array, called M31hat(5).
%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(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 Fourth member (K=5) in the family M31hat(K) of partition number arrays.
%C If M31hat(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1hat(5):= A144891.
%H W. Lang, <a href="/A144890/a144890.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(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)| = A049353(n,1) = A001720(n+3) = [1,5,30,210,1680,...] = (n+3)!/4!, 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];[5,1];[30,5,1];[210,30,25,5,1];[1680,210,150,30,25,5,1];...
%e a(4,3)= 25 = |S1(5;2,1)|^2. The relevant partition of 4 is (2^2).
%Y Cf. A144892 (row sums).
%Y Cf. A144885 (M31hat(4) array). A144891 (S1hat(5).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008
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