A144877 Partition number array, called M31(-3), related to A049410(n,m) = S1(-3;n,m) (generalized Stirling triangle).

1, 3, 1, 6, 9, 1, 6, 24, 27, 18, 1, 0, 30, 180, 60, 135, 30, 1, 0, 0, 270, 360, 90, 1080, 405, 120, 405, 45, 1, 0, 0, 0, 1260, 0, 1890, 2520, 5670, 210, 3780, 2835, 210, 945, 63, 1, 0, 0, 0, 0, 1260, 0, 0, 10080, 11340, 30240, 0, 7560, 10080, 45360, 8505, 420, 10080, 11340

Offset: 1

Author

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Keywords

• nonn
• easy
• tabf

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(-3;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, ...].

First member (K=3) in the family M31(-K) of partition number arrays.

If M31(-3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-3) := A049410.

Examples

			[1]; [3,1]; [6,9,1]; [6,24,27,18,1]; [0,30,180,60,135,30,1]; ...
a(4,3) = 27 = 3*S1(-3;2,1)^2. The relevant partition of 4 is (2^2).
		

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.

Crossrefs

Cf. A049426 (row sums).

Cf. A144358 (M31(-2) array), A144878 (M31(-4) array).

Formula

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(-3;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-3;j,1)^e(n,k,j),j=1..n) with S1(-3;n,1)|= A008279(3,n-1)= [1,3,6,6,0,...], 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.