A144269 - OEIS

%I #13 Jul 01 2023 12:11:58

%S 1,1,1,3,1,1,15,3,1,1,1,105,15,3,3,1,1,1,945,105,15,9,15,3,1,3,1,1,1,

%T 10395,945,105,45,105,15,9,3,15,3,1,3,1,1,1,135135,10395,945,315,225,

%U 945,105,45,15,9,105,15,9,3,1,15,3,1,3,1,1,1,2027025,135135,10395,2835

%N Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-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) =: M32hat(-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 If M32hat(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-1):= A144270(n,m).

%H Wolfdieter Lang, <a href="/A144269/a144269.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)= product(|S2(-1,j,1)|^e(n,k,j),j=1..n) with |S2(-1,n,1)|= A001147(n-1) = (2*n-3)(!^2) (2-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(-1)/M3' = 'A143171/A036040' (elementwise division of arrays).

%e a(4,3)= 1 = |S2(-1,2,1)|^2. The relevant partition of 4 is (2^2).

%e [1]; [1,1]; [3,1,1]; [15,3,1,1,1]; [105,15,3,3,1,1,1]; ... [From _Wolfdieter Lang_, Oct 23 2008]

%Y Cf. A144271 (M32hat(-2) array).

%K nonn,easy,tabf

%O 1,4

%A _Wolfdieter Lang_, Oct 09 2008

%E Corrected all entries. _Wolfdieter Lang_, Oct 23 2008