A144888 -id:A144888 - cp's OEIS frontend

A144886 Lower triangular array called S1hat(4) related to partition number array A144885.

1, 4, 1, 20, 4, 1, 120, 36, 4, 1, 840, 200, 36, 4, 1, 6720, 1720, 264, 36, 4, 1, 60480, 12480, 2040, 264, 36, 4, 1, 604800, 118560, 16000, 2296, 264, 36, 4, 1, 6652800, 1081920, 149600, 17280, 2296, 264, 36, 4, 1, 79833600, 11793600, 1362240, 163680, 18304, 2296, 264

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M31hat(4):=A144885 entries with the same parts number m are summed one obtains this triangle of numbers S1hat(4). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first columns are A001715(n+2), A144888, A144889,...

			[1];[4,1];[20,4,1];[120,36,4,1];[840,200,36,4,1];...

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.

A144887 (row sums).

a(n,m)=sum(product(|S1(4;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(4,n,1)|= A049352(n,1) = A001715(n+2) = (n+2)!/3!.

A144889 Third column (m=3) of triangle A144886 (S1hat(4)).

Original entry on oeis.org

1, 4, 36, 264, 2040, 16000, 149600, 1362240, 14326080, 158438400, 1931731200, 25155648000, 357727910400, 5420614348800, 88408793088000, 1532112261120000, 28238983686144000, 549825606070272000, 11292931674759168000, 243644152971018240000, 5511211980392079360000

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144888 (second column).

a(n) = A144886(n+3,3), n>=0.

cp's OEIS Frontend

A144886 Lower triangular array called S1hat(4) related to partition number array A144885.

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