A144883 -id:A144883 - cp's OEIS frontend

A144881 Lower triangular array called S1hat(3) related to partition number array A144880.

1, 3, 1, 12, 3, 1, 60, 21, 3, 1, 360, 96, 21, 3, 1, 2520, 684, 123, 21, 3, 1, 20160, 4320, 792, 123, 21, 3, 1, 181440, 35640, 5292, 873, 123, 21, 3, 1, 1814400, 293760, 42768, 5616, 873, 123, 21, 3, 1, 19958400, 2881440, 348840, 45684, 5859, 873, 123, 21, 3, 1, 239500800

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M31hat(3):=A144880 entries 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 columns are A001710(n+1), A144883, A144884,...

			[1];[3,1];[12,3,1];[60,21,3,1];[360,96,21,3,1];...

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

A144882 (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)|= A046089(n,1) = A001710(n+1) = (n+1)!/2.

A144884 Third column (m=3) of triangle A144881 (S1hat(3)).

Original entry on oeis.org

1, 3, 21, 123, 792, 5292, 42768, 348840, 3313440, 33734880, 382423680, 4683303360, 62795865600, 903908678400, 14018003596800, 231803919244800, 4081121327001600, 76062618390528000, 1497060883196928000, 31001124301830144000, 673857550403702784000

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144881, A144883 (second column).

a(n) = A144881(n+3,3), n>=0.

cp's OEIS Frontend

A144881 Lower triangular array called S1hat(3) related to partition number array A144880.

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