A144878 Partition number array, called M31(-4), related to A049424(n,m) = S1(-4;n,m) (generalized Stirling triangle).

1, 4, 1, 12, 12, 1, 24, 48, 48, 24, 1, 24, 120, 480, 120, 240, 40, 1, 0, 144, 1440, 1440, 360, 2880, 960, 240, 720, 60, 1, 0, 0, 2016, 10080, 504, 10080, 10080, 20160, 840, 10080, 6720, 420, 1680, 84, 1, 0, 0, 0, 16128, 20160, 0, 16128, 80640, 80640, 161280, 1344, 40320

Offset

1,2

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(-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, ...].

First member (K=4) in the family M31(-K) of partition number arrays.

If M31(-4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-4) := A049424.

Links

Table of n, a(n) for n=1..56.

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)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(-4;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-4;j,1)^e(n,k,j),j=1..n) with S1(-4;n,1)= A008279(4,n-1)= [1,4,12,12,24,24,0,...], 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.

Example

[1]; [4,1]; [12,12,1]; [24,48,48,24,1]; [24,120,480,120,240,40,1]; ...
a(4,3) = 48 = 3*S1(-4;2,1)^2. The relevant partition of 4 is (2^2).

Crossrefs

Cf. A049427 (row sums).

Cf. A144877 (M31(-3) array), A144879 (M31(-5) array).

Sequence in context: A051290 A227338 A125105 * A049424 A157394 A338864

Adjacent sequences: A144875 A144876 A144877 * A144879 A144880 A144881

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang Oct 09 2008, Oct 28 2008

Status

approved