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%I #20 Aug 29 2019 16:44:58
%S 1,1,1,0,3,1,0,0,3,6,1,0,0,0,0,15,10,1,0,0,0,0,0,0,15,0,45,15,1,0,0,0,
%T 0,0,0,0,0,0,0,105,0,105,21,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,105,0,0,420,
%U 0,210,28,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,945,0,0,1260,0,378,36
%N Partition number array, called M31(-1), related to A049403(n,m) = S1(-1;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(-1;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=1) in the family M31(-K) of partition number arrays.
%C If M31(-1;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-1) := A049403.
%H Wolfdieter Lang, <a href="/A144357/a144357.txt">First 10 rows of the array and more</a>.
%H Wolfdieter 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(-1;j,1)^e(n,k,j) = M3(n,k)*Product_{j=1..n} S1(-1;j,1)^e(n,k,j) with S1(-1;n,1) |= A008279(1,n-1) = [1,1,0,...], 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]; [1,1]; [0,3,1]; [0,0,3,6,1]; [0,0,0,0,15,10,1]; ...
%e a(4,3) = 3 = 3*S1(-1;2,1)^2. The relevant partition of 4 is (2^2).
%Y Cf. A000085 (row sums).
%Y Cf. A144358 (M31(-2) array).
%K nonn,easy,tabf
%O 1,5
%A _Wolfdieter Lang_, Oct 09 2008, Oct 28 2008
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