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

Author

Wolfdieter Lang, Oct 17 2008, Oct 28 2008

Keywords

• nonn
• easy
• tabf

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.

Examples

			Triangle begins
  [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).
		

Links

• Wolfdieter Lang, First 10 rows of the array and more.
• Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Crossrefs

Cf. A145358 (row sums).

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

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.