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A144284 Partition number array, called M32hat(-4)= 'M32(-4)/M3'= 'A144267/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

%I A144284 #9 Aug 29 2019 16:42:58
%S A144284 1,4,1,36,4,1,504,36,16,4,1,9576,504,144,36,16,4,1,229824,9576,2016,
%T A144284 1296,504,144,64,36,16,4,1,6664896,229824,38304,18144,9576,2016,1296,
%U A144284 576,504,144,64,36,16,4,1,226606464,6664896,919296,344736,254016,229824,38304,18144
%N A144284 Partition number array, called M32hat(-4)= 'M32(-4)/M3'= 'A144267/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).
%C A144284 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) =: M32hat(-4;n,k) with the k-th partition of n in A-St order.
%C A144284 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A144284 If M32hat(-4;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-4):= A144285(n,m).
%H A144284 W. Lang, <a href="/A144284/a144284.txt">First 10 rows of the array and more.</a>
%H A144284 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 A144284 a(n,k)= product(|S2(-4,j,1)|^e(n,k,j),j=1..n) with |S2(-4,n,1)|= A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if 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.
%F A144284 Formally a(n,k)= 'M32(-4)/M3' = 'A144267/A036040' (elementwise division of arrays).
%e A144284 a(4,3)= 16 = |S2(-4,2,1)|^2. The relevant partition of 4 is (2^2).
%Y A144284 A144279 (M32hat(-3) array). A144341 (M32hat(-5) array)
%K A144284 nonn,easy,tabf
%O A144284 1,2
%A A144284 _Wolfdieter Lang_ Oct 09 2008