A145373 -id:A145373 - cp's OEIS frontend

A145374 Row sums of triangle A145373 (S1hat(-5)) and partition array A145372 (M31hat(-5)).

1, 6, 26, 111, 331, 1276, 3576, 14301, 43401, 142626, 422526, 1501051, 4244551, 13621176, 41578676, 130893801, 378009301, 1188872926, 3428610426, 10769968551, 31120656051, 92887286676, 269920084176, 830429397301, 2353647784801

Offset: 0

Wolfdieter Lang, Oct 17 2008, Nov 13 2008

Cf. A145372, A145373, A145371 (row sums of triangle A145370 (S1hat(-4))).

a(n) = Sum_{m=1..n} A145373(n,m), n>=1.

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

Original entry on oeis.org

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

Wolfdieter Lang, Oct 17 2008

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.

			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).

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.

Cf. A145369 (M31hat(-4)).

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.

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A145374 Row sums of triangle A145373 (S1hat(-5)) and partition array A145372 (M31hat(-5)).

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

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