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%I #13 Aug 29 2019 15:53:54
%S 1,2,1,2,6,1,0,8,12,12,1,0,0,40,20,60,20,1,0,0,0,40,0,240,120,40,180,
%T 30,1,0,0,0,0,0,0,280,840,0,840,840,70,420,42,1,0,0,0,0,0,0,0,0,0,
%U 2240,0,0,1120,6720,1680,0,2240,3360,112,840,56,1,0,0,0,0,0,0,0,0,0,0,0,2240,0,0
%N Partition number array, called M31(-2), related to A049404(n,m) = S1(-2;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(-2;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=2) in the family M31(-K) of partition number arrays.
%C If M31(-2;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-2) := A049404.
%H W. Lang, <a href="/A144358/a144358.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(-2;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-2;j,1)^e(n,k,j),j=1..n) with S1(-2;n,1)|= A008279(2,n-1)= [1,2,2,0,...], 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]; [2,1]; [2,6,1]; [0,8,12,12,1]; [0,0,40,20,60,20,1]; ...
%e a(4,3) = 12 = 3*S1(-2;2,1)^2. The relevant partition of 4 is (2^2).
%Y Cf. A049425 (row sums).
%Y Cf. A144357 (M31(-1) array), A144877 (M31(-3) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008
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