A145369 -id:A145369 - cp's OEIS frontend

A145370 Lower triangular array, called S1hat(-4), related to partition number array A145369.

1, 4, 1, 12, 4, 1, 24, 28, 4, 1, 24, 72, 28, 4, 1, 0, 264, 136, 28, 4, 1, 0, 384, 456, 136, 28, 4, 1, 0, 864, 1344, 712, 136, 28, 4, 1, 0, 576, 4128, 2112, 712, 136, 28, 4, 1, 0, 576, 7488, 7968, 3136, 712, 136, 28, 4, 1, 0, 0, 13248, 20544, 11040, 3136, 712, 136, 28, 4, 1, 0, 0

Offset: 1

Wolfdieter Lang, Oct 17 2008

If in the partition array M31hat(-4):=A145369 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-4). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first column is [1,4,12,24,24,0,0,0,...]= A008279(4,n-1), n>=1.

			Triangle begins:
  [1];
  [4,1];
  [12,4,1];
  [24,28,4,1];
  [24,72,28,4,1];
  ...

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. A145371 (row sums).

a(n,m) = sum(product(S1(-4;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n, Y and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S1(-4,n,1)= A008279(4,n-1) = [1,4,12,24,24,0,0,0,...], n>=1.

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

Original entry on oeis.org

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

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 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.

			Triangle begins:
  [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).

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

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.

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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A145370 Lower triangular array, called S1hat(-4), related to partition number array A145369.

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

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

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