A145369 Partition number array, called M31hat(-4).

1, 4, 1, 12, 4, 1, 24, 12, 16, 4, 1, 24, 24, 48, 12, 16, 4, 1, 0, 24, 96, 144, 24, 48, 64, 12, 16, 4, 1, 0, 0, 96, 288, 24, 96, 144, 192, 24, 48, 64, 12, 16, 4, 1, 0, 0, 0, 288, 576, 0, 96, 288, 384, 576, 24, 96, 144, 192, 256, 24, 48, 64, 12, 16, 4, 1, 0, 0, 0, 0, 576, 0, 0, 288, 576, 384

Offset

1,2

Comments

If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2,3,4 or 5 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).

Fourth member (K=4) in the family M31hat(-K) of partition number arrays.

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].

This array is array A144878 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144878/A036040'. E.g. a(4,3)= 16 = 48/3 = A144878(4,3)/A036040(4,3).

If M31hat(-4;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-4):= A145370.

Links

Table of n, a(n) for n=1..76.

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

Example

[1];[4,1];[12,4,1];[24,12,16,4,1];[24,24,48,12,16,4,1];...
a(4,3)= 16 = S1(-4;2,1)^2. The relevant partition of 4 is (2^2).

Crossrefs

A145366 (M31hat(-3)). A145372 (M31hat(-5)).

Sequence in context: A078710 A175763 A080303 * A145370 A130322 A316232

Adjacent sequences: A145366 A145367 A145368 * A145370 A145371 A145372

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang Oct 17 2008

Status

approved