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A144353 Partition number array, called M31(3), related to A046089(n,m)= |S1(3;n,m)| (generalized Stirling triangle).

%I A144353 #10 Aug 29 2019 17:52:31
%S A144353 1,3,1,12,9,1,60,48,27,18,1,360,300,360,120,135,30,1,2520,2160,2700,
%T A144353 1440,900,2160,405,240,405,45,1,20160,17640,22680,25200,7560,18900,
%U A144353 10080,11340,2100,7560,2835,420,945,63,1,181440,161280,211680,241920,126000,70560,181440
%N A144353 Partition number array, called M31(3), related to A046089(n,m)= |S1(3;n,m)| (generalized Stirling triangle).
%C A144353 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(3;n,k) with the k-th partition of n in A-St order.
%C A144353 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A144353 Third member (K=3) in the family M31(K) of partition number arrays.
%C A144353 If M31(3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(3)|:= A046089.
%H A144353 W. Lang, <a href="/A144353/a144353.txt">First 10 rows of the array and more.</a>
%H A144353 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 A144353 a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(3;j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S1(3;j,1)|^e(n,k,j),j=1..n) with |S1(3;n,1)|= A001710(n+1) = (n+1)!/2!, 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 A144353 [1];[3,1];[12,9,1];[60,48,27,18,1];[360,300,360,120,135,30,1];...
%e A144353 a(4,3)= 27 = 3*|S1(3;2,1)|^2. The relevant partition of 4 is (2^2).
%Y A144353 A049376 (row sums).
%Y A144353 A130561 (M31(2) array), A144354 (M31(4) array).
%K A144353 nonn,easy,tabf
%O A144353 1,2
%A A144353 _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008