A145368 -id:A145368 - 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.

A145371 Row sums of triangle A145370 (S1hat(-4)) and partition array A145359 (M31hat(-4)).

Original entry on oeis.org

1, 5, 17, 57, 129, 433, 1009, 3089, 7697, 20049, 48849, 139729, 317137, 818897, 2040017, 5125329, 12000465, 30811345, 71582929, 179008721, 427693265, 1024611537, 2413499601, 6027416785, 13788615889

Offset: 1

Wolfdieter Lang, Oct 17 2008

Cf. A145359, A145370, A145368 (row sums of triangle A145367).

a(n) = Sum_{m=1..n} A145370(n,m), n>=1.

cp's OEIS Frontend

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

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