OFFSET
1,2
COMMENTS
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) =: M31(-3;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
First member (K=3) in the family M31(-K) of partition number arrays.
If M31(-3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-3) := A049410.
LINKS
W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
FORMULA
a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(-3;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-3;j,1)^e(n,k,j),j=1..n) with S1(-3;n,1)|= A008279(3,n-1)= [1,3,6,6,0,...], 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. M3(n,k)=A036040.
EXAMPLE
[1]; [3,1]; [6,9,1]; [6,24,27,18,1]; [0,30,180,60,135,30,1]; ...
a(4,3) = 27 = 3*S1(-3;2,1)^2. The relevant partition of 4 is (2^2).
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang Oct 09 2008, Oct 28 2008
STATUS
approved