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A144274 Partition number array, called M32hat(-2)= 'M32(-2)/M3'= 'A143172/A036040', related to A004747(n,m)= |S2(-2;n,m)| (generalized Stirling triangle).

%I A144274 #16 Jul 02 2023 09:01:46
%S A144274 1,2,1,10,2,1,80,10,4,2,1,880,80,20,10,4,2,1,12320,880,160,100,80,20,
%T A144274 8,10,4,2,1,209440,12320,1760,800,880,160,100,40,80,20,8,10,4,2,1,
%U A144274 4188800,209440,24640,8800,6400,12320,1760,800,320,200,880,160,100,40,16,80,20
%N A144274 Partition number array, called M32hat(-2)= 'M32(-2)/M3'= 'A143172/A036040', related to A004747(n,m)= |S2(-2;n,m)| (generalized Stirling triangle).
%C A144274 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(-2;n,k) with the k-th partition of n in A-St order.
%C A144274 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A144274 If M32hat(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-2):= A144275(n,m).
%H A144274 Wolfdieter Lang, <a href="/A144274/a144274.txt">First 10 rows of the array and more.</a>
%H A144274 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 A144274 a(n,k) = Product_{j=1..n} |S2(-2,j,1)|^e(n,k,j) with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-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 A144274 Formally a(n,k)= 'M32(-2)/M3' = 'A143172/A036040' (elementwise division of arrays).
%e A144274 a(4,3) = 4 = |S2(-2,2,1)|^2. The relevant partition of 4 is (2^2).
%Y A144274 Cf. A144269 (M32hat(-1) array). A144279 (M32hat(-3) array).
%K A144274 nonn,easy,tabf
%O A144274 1,2
%A A144274 _Wolfdieter Lang_, Oct 09 2008