A145366 Partition number array, called M31hat(-3).

1, 3, 1, 6, 3, 1, 6, 6, 9, 3, 1, 0, 6, 18, 6, 9, 3, 1, 0, 0, 18, 36, 6, 18, 27, 6, 9, 3, 1, 0, 0, 0, 36, 0, 18, 36, 54, 6, 18, 27, 6, 9, 3, 1, 0, 0, 0, 0, 36, 0, 0, 36, 54, 108, 0, 18, 36, 54, 81, 6, 18, 27, 6, 9, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 108, 216, 0, 0, 36, 54, 108, 162, 0, 18, 36, 54, 81

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 or 4 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).

Third member (K=3) 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 A144877 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144877/A036040'. E.g. a(4,3)= 9 = 27/3 = A144877(4,3)/A036040(4,3).

If M31hat(-3;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-3):= A145367.

Links

Table of n, a(n) for n=1..89.

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(-3;j,1)^e(n,k,j),j=1..n) with S1(-3;n,1) = A008279(3,n-1) = [1,3,6,6,0,0,0,...], 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];[3,1];[6,3,1];[6,6,9,3,1];[0,6,18,6,9,3,1];...
a(4,3)= 9 = S1(-3;2,1)^2. The relevant partition of 4 is (2^2).

Crossrefs

A145363 (M31hat(-2)). A145369 (M31hat(-4))

Sequence in context: A336173 A072361 A341219 * A145367 A124928 A249250

Adjacent sequences: A145363 A145364 A145365 * A145367 A145368 A145369

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang Oct 17 2008

Status

approved