A144269 -id:A144269 - cp's OEIS frontend

A144270 Lower triangular array called S2hat(-1) related to partition number array A144269.

1, 1, 1, 3, 1, 1, 15, 4, 1, 1, 105, 18, 4, 1, 1, 945, 129, 19, 4, 1, 1, 10395, 1095, 132, 19, 4, 1, 1, 135135, 11880, 1119, 133, 19, 4, 1, 1, 2027025, 149940, 12057, 1122, 133, 19, 4, 1, 1, 34459425, 2218545, 151560, 12081, 1123, 133, 19, 4, 1, 1

Offset: 1

Wolfdieter Lang, Oct 09 2008

If in the partition array M32hat(-1)=A144269 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-1). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.

The first three columns are A001147, A144272, A144273.

			Triangle begins
    1;
    1,  1;
    3,  1, 1;
   15,  4, 1, 1;
  105, 18, 4, 1, 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.

Row sums A144271.

a(n,m)=sum(product(|S2(-1;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. |S2(-1;j,1)|= A001497(j-1,0) = A001147(j-1) = (2*j-3)(!^2) (2-factorials) for j>=2 and 1 if j=1.

A144274 Partition number array, called M32hat(-2)= 'M32(-2)/M3'= 'A143172/A036040', related to A004747(n,m)= |S2(-2;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 2, 1, 10, 2, 1, 80, 10, 4, 2, 1, 880, 80, 20, 10, 4, 2, 1, 12320, 880, 160, 100, 80, 20, 8, 10, 4, 2, 1, 209440, 12320, 1760, 800, 880, 160, 100, 40, 80, 20, 8, 10, 4, 2, 1, 4188800, 209440, 24640, 8800, 6400, 12320, 1760, 800, 320, 200, 880, 160, 100, 40, 16, 80, 20

Offset: 1

Wolfdieter Lang, Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M32hat(-2;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
If M32hat(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-2):= A144275(n,m).

			a(4,3) = 4 = |S2(-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. A144269 (M32hat(-1) array). A144279 (M32hat(-3) array).

a(n,k) = Product_{j=1..n} |S2(-2,j,1)|^e(n,k,j) with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-factorials) for n>=2 and 1 if 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.

Formally a(n,k)= 'M32(-2)/M3' = 'A143172/A036040' (elementwise division of arrays).

cp's OEIS Frontend

A144270 Lower triangular array called S2hat(-1) related to partition number array A144269.

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