A144279 - OEIS

%I #13 Jul 02 2023 12:47:55

%S 1,3,1,21,3,1,231,21,9,3,1,3465,231,63,21,9,3,1,65835,3465,693,441,

%T 231,63,27,21,9,3,1,1514205,65835,10395,4851,3465,693,441,189,231,63,

%U 27,21,9,3,1,40883535,1514205,197505,72765,53361,65835,10395,4851,2079,1323,3465

%N Partition number array, called M32hat(-3)= 'M32(-3)/M3'= 'A143173/A036040', related to A000369(n,m)= |S2(-3;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(-3;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(-3;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-3):= A144280(n,m).

%H Wolfdieter Lang, <a href="/A144279/a144279.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_{j=1..n} |S2(-3,j,1)|^e(n,k,j), with |S2(-3,n,1)|= A008545(n-1) = (4*n-5)(!^4) (4-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(-3)/M3' = 'A143173/A036040' (elementwise division of arrays).

%e a(4,3) = 9 = |S2(-3,2,1)|^2. The relevant partition of 4 is (2^2).

%Y Cf. A036040, A143173, A134278, A000041, A008545.

%Y Cf. A144274 (M32hat(-2) array), A144284 (M32hat(-4) array).

%K nonn,easy,tabf

%O 1,2

%A _Wolfdieter Lang_, Oct 09 2008