A145356 Partition number array, called M31hat(6).

1, 6, 1, 42, 6, 1, 336, 42, 36, 6, 1, 3024, 336, 252, 42, 36, 6, 1, 30240, 3024, 2016, 1764, 336, 252, 216, 42, 36, 6, 1, 332640, 30240, 18144, 14112, 3024, 2016, 1764, 1512, 336, 252, 216, 42, 36, 6, 1, 3991680, 332640, 181440, 127008, 112896, 30240, 18144, 14112

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..52.

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(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.

Example

[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).

Crossrefs

Cf. A145358 (row sums).

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

Sequence in context: A145927 A113365 A293172 * A145357 A035529 A135893

Adjacent sequences: A145353 A145354 A145355 * A145357 A145358 A145359

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang Oct 17 2008, Oct 28 2008

Status

approved