A145366 -id:A145366 - cp's OEIS frontend

A145367 Lower triangular array, called S1hat(-3), related to partition number array A145366.

1, 3, 1, 6, 3, 1, 6, 15, 3, 1, 0, 24, 15, 3, 1, 0, 54, 51, 15, 3, 1, 0, 36, 108, 51, 15, 3, 1, 0, 36, 198, 189, 51, 15, 3, 1, 0, 0, 360, 360, 189, 51, 15, 3, 1, 0, 0, 324, 846, 603, 189, 51, 15, 3, 1, 0, 0, 216, 1296, 1332, 603, 189, 51, 15, 3, 1, 0, 0, 216, 2484, 2754, 2061, 603, 189

Offset: 1

Wolfdieter Lang, Oct 17 2008

If in the partition array M31hat(-3):=A145366 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-3). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first column is [1,3,6,6,0,0,0,...]= A008279(3,n-1), n>=1.

			Triangle begins:
  [1];
  [3,1];
  [6,3,1];
  [6,15,3,1];
  [0,24,15,3,1];
  ...

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. A145368 (row sums).

a(n,m) = sum(product(S1(-3;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 and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S1(-3,n,1)= A008279(3,n-1) = [1,3,6,6,0,0,0,...], n>=1.

A145368 Row sums of triangle A145367 (S1hat(-3)) and partition array A145366 (M31hat(-3)).

Original entry on oeis.org

1, 4, 10, 25, 43, 124, 214, 493, 979, 2032, 3706, 8377, 14695, 30004, 58030, 113029, 204883, 424312, 754306, 1467025, 2783599, 5248348, 9524662, 18925117, 33713611

Offset: 0

Wolfdieter Lang, Oct 17 2008

Cf. A145367, A145366, A145365 (row sums of triangle A145364).

a(n) = Sum_{m=1..n} A145367(n,m), for n>=1.

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

Original entry on oeis.org

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

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 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.

			Triangle begins
  [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).

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

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.

A145369 Partition number array, called M31hat(-4).

Original entry on oeis.org

1, 4, 1, 12, 4, 1, 24, 12, 16, 4, 1, 24, 24, 48, 12, 16, 4, 1, 0, 24, 96, 144, 24, 48, 64, 12, 16, 4, 1, 0, 0, 96, 288, 24, 96, 144, 192, 24, 48, 64, 12, 16, 4, 1, 0, 0, 0, 288, 576, 0, 96, 288, 384, 576, 24, 96, 144, 192, 256, 24, 48, 64, 12, 16, 4, 1, 0, 0, 0, 0, 576, 0, 0, 288, 576, 384

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 or 5 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
Fourth member (K=4) 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 A144878 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144878/A036040'. E.g. a(4,3)= 16 = 48/3 = A144878(4,3)/A036040(4,3).
If M31hat(-4;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-4):= A145370.

			Triangle begins:
  [1];
  [4,1];
  [12,4,1];
  [24,12,16,4,1];
  [24,24,48,12,16,4,1];
  ...
a(4,3)= 16 = 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. A145366 (M31hat(-3)), A145372 (M31hat(-5)).

a(n,k) = product(S1(-4;j,1)^e(n,k,j),j=1..n) with S1(-4;n,1) = A008279(4,n-1) = [1,4,12,24,24,0,0,0,...], 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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A145367 Lower triangular array, called S1hat(-3), related to partition number array A145366.

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A145368 Row sums of triangle A145367 (S1hat(-3)) and partition array A145366 (M31hat(-3)).

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

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A145369 Partition number array, called M31hat(-4).

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