A145372 Partition number array, called M31hat(-5).

1, 5, 1, 20, 5, 1, 60, 20, 25, 5, 1, 120, 60, 100, 20, 25, 5, 1, 120, 120, 300, 400, 60, 100, 125, 20, 25, 5, 1, 0, 120, 600, 1200, 120, 300, 400, 500, 60, 100, 125, 20, 25, 5, 1, 0, 0, 600, 2400, 3600, 120, 600, 1200, 1500, 2000, 120, 300, 400, 500, 625, 60, 100, 125, 20

Offset: 1

Author

Wolfdieter Lang, Oct 17 2008

Keywords

• nonn
• easy
• tabf

Comments

If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2,3,4,5 or 6 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).

Fifth member (K=5) 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 A144879 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144879/A036040'. E.g. a(4,3)= 25 = 75/3 = A144879(4,3)/A036040(4,3).

If M31hat(-5;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-5):= A145373.

Examples

			Triangle begins;
  [1];
  [5,1];
  [20,5,1];
  [60,20,25,5,1];
  [120,60,100,20,25,5,1];
  ...
a(4,3)= 25 = S1(-4;2,1)^2. The relevant partition of 4 is (2^2).
		

Links

• Wolfdieter Lang, First 10 rows of the array and more.
• Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Crossrefs

Cf. A145369 (M31hat(-4)).

Formula

a(n,k) = product(S1(-5;j,1)^e(n,k,j),j=1..n) with S1(-5;n,1) = A008279(5,n-1) = [1,5,20,60,120,120,0,0,0,...], 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.