A144885 - OEIS

%I #10 Aug 29 2019 17:12:27

%S 1,4,1,20,4,1,120,20,16,4,1,840,120,80,20,16,4,1,6720,840,480,400,120,

%T 80,64,20,16,4,1,60480,6720,3360,2400,840,480,400,320,120,80,64,20,16,

%U 4,1,604800,60480,26880,16800,14400,6720,3360,2400,1920,1600,840,480,400,320

%N Partition number array, called M31hat(4).

%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) =: M31hat(4;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 Fourth member (K=4) in the family M31hat(K) of partition number arrays.

%C If M31hat(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1hat(4):= A144886.

%H W. Lang, <a href="/A144885/a144885.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(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)| = A049352(n,1) = A001715(n+2) = [1,4,20,120,840,6720,...] = (n+2)!/3!, 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.

%e [1];[4,1];[20,4,1];[120,20,16,4,1];[840,120,80,20,16,4,1];...

%e a(4,3)= 16 = |S1(4;2,1)|^2. The relevant partition of 4 is (2^2).

%Y A144887 (row sums).

%Y A144880 (M31hat(3) array). A144886 (S1hat(4)).

%K nonn,easy,tabf

%O 1,2

%A _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008