OFFSET
1,2
COMMENTS
If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2,3 or 4 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
Third member (K=3) in the family M31hat(-K) of partition number arrays.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
This array is array A144877 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144877/A036040'. E.g. a(4,3)= 9 = 27/3 = A144877(4,3)/A036040(4,3).
If M31hat(-3;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-3):= A145367.
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) = product(S1(-3;j,1)^e(n,k,j),j=1..n) with S1(-3;n,1) = A008279(3,n-1) = [1,3,6,6,0,0,0,...], 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.
EXAMPLE
[1];[3,1];[6,3,1];[6,6,9,3,1];[0,6,18,6,9,3,1];...
a(4,3)= 9 = S1(-3;2,1)^2. The relevant partition of 4 is (2^2).
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang Oct 17 2008
STATUS
approved