A145363 Partition number array, called M31hat(-2).

1, 2, 1, 2, 2, 1, 0, 2, 4, 2, 1, 0, 0, 4, 2, 4, 2, 1, 0, 0, 0, 4, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 4, 8, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 4, 8, 16, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 8, 16, 0, 0, 4, 8, 16, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,2

Comments

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

Second member (K=2) 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 A144358 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144358/A036040'. E.g. a(4,3)= 4 = 12/3 = A144358(4,3)/A036040(4,3).

If M31hat(-2;n,k) is summed over those k belonging to partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-2):= A145364.

Links

Table of n, a(n) for n=1..104.

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(-2;j,1)^e(n,k,j),j=1..n) with S1(-2;n,1) = A008279(2,n-1) = [1,2,1,0,0,0,...], 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];[2,1];[2,2,1];[0,2,4,2,1];[0,0,4,2,4,2,1];...
a(4,3)= 4 = S1(-2;2,1)^2. The relevant partition of 4 is (2^2).

Crossrefs

A145361 (M31hat(-1)). A145366 (M31hat(-3))

Sequence in context: A060208 A004570 A178064 * A071429 A264398 A145364

Adjacent sequences: A145360 A145361 A145362 * A145364 A145365 A145366

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang Oct 17 2008

Status

approved