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(5;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,...].
Fifth member (K=5) in the family M31(K) of partition number arrays.
If M31(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(5)|:= A049353.
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(5;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)|= A001720(n+3) = (n+3)!/4!, 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];[5,1];[30,15,1];[210,120,75,30,1];[1680,1050,1500,300,375,50,1];...
a(4,3)= 75 = 3*|S1(5;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