A144894 -id:A144894 - cp's OEIS frontend

A144891 Lower triangular array called S1hat(5) related to partition number array A144890.

1, 5, 1, 30, 5, 1, 210, 55, 5, 1, 1680, 360, 55, 5, 1, 15120, 3630, 485, 55, 5, 1, 151200, 29820, 4380, 485, 55, 5, 1, 1663200, 321300, 39570, 5005, 485, 55, 5, 1, 19958400, 3225600, 421800, 43320, 5005, 485, 55, 5, 1, 259459200, 38808000, 4265100, 470550, 46445, 5005

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M31hat(5):=A144890 entries with the same parts number m are summed one obtains this triangle of numbers S1hat(5). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first columns are A001720(n+3)=(n+3)!/4!, A144893, A144894,...

			[1];[5,1];[30,5,1];[210,55,5,1];[1680,360,55,5,1];...

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.

A144892 (row sums).

a(n,m)=sum(product(|S1(5;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(5,n,1)|= A049353(n,1) = A001720(n+3) = (n+3)!/4!.

A144893 Second column (m=2) of triangle A144891 (S1hat(5)).

Original entry on oeis.org

1, 5, 55, 360, 3630, 29820, 321300, 3225600, 38808000, 466300800, 6360379200, 89703936000, 1389213504000, 22565765376000, 394272204480000, 7248941973504000, 141496402037760000, 2901258659819520000, 62617333274496000000, 1414755114367795200000, 33446554797269053440000

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144891, A001720 (first column), A144894 (third column).

a(n) = A144891(n+2,2), n>=0.

cp's OEIS Frontend

A144891 Lower triangular array called S1hat(5) related to partition number array A144890.

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