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

1,4

Comments

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.

Links

Table of n, a(n) for n=1..55.

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.

Formula

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.

Example

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

Crossrefs

Row sums A144271.

Sequence in context: A006956 A072285 A144269 * A110112 A326800 A176225

Adjacent sequences: A144267 A144268 A144269 * A144271 A144272 A144273

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Oct 09 2008

Status

approved