|
%I #10 Aug 29 2019 17:51:47
%S 1,5,1,55,5,1,935,55,25,5,1,21505,935,275,55,25,5,1,623645,21505,4675,
%T 3025,935,275,125,55,25,5,1,21827575,623645,107525,51425,21505,4675,
%U 3025,1375,935,275,125,55,25,5,1,894930575,21827575,3118225,1182775,874225,623645
%N Partition number array, called M32hat(-5)= 'M32(-5)/M3'= 'A144268/A036040', related to A011801(n,m)= |S2(-4;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) =: M32hat(-5;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 If M32hat(-5;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-5):= A144342(n,m).
%H W. Lang, <a href="/A144341/a144341.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)= product(|S2(-5,j,1)|^e(n,k,j),j=1..n) with |S2(-5,n,1)|= A008543(n-1) = (6*n-7)(!^6) (6-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 Formally a(n,k)= 'M32(-5)/M3' = 'A144268/A036040' (elementwise division of arrays).
%e a(4,3)= 25 = |S2(-5,2,1)|^2. The relevant partition of 4 is (2^2).
%Y A144284 (M32hat(-4) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_ Oct 09 2008
|
