1,2

Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(6;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,...].

Sixth member (K=6) in the family M31hat(K) of partition number arrays.

If M31hat(6;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(6):= A145357.

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(6;j,1)|^e(n,k,j),j=1..n) with |S1(6;n,1)| = A049374(n,1) = A001725(n+4) = [1,6,42,336,3024,30240,332640,...] = (n+4)!/5!, 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];[6,1];[42,6,1];[336,42,36,6,1];[3024,336,252,42,36,6,1];...

a(4,3)= 36 = |S1(6;2,1)|^2. The relevant partition of 4 is (2^2).

A145358 (row sums).

A144890 (M31hat(5) array). A145357 (S1hat(6).

Sequence in context: A089504 A145927 A113365 this_sequence A145357 A035529 A135893

Adjacent sequences: A145353 A145354 A145355 this_sequence A145357 A145358 A145359

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 17 2008, Oct 28 2008