1,2

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.

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].

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

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

W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, preprint Oct 2008.

W. Lang, First 10 rows of the array and more.

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.

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

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

A144887 (row sums).

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

Sequence in context: A141233 A055139 A128041 this_sequence A144886 A117380 A167432

Adjacent sequences: A144882 A144883 A144884 this_sequence A144886 A144887 A144888

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 09 2008, Oct 28 2008