A144285 -id:A144285 - cp's OEIS frontend

A144339 Second column (m=2) of triangle S2hat(-4) = A144285.

1, 4, 52, 648, 12888, 286272, 8182944, 266366016, 10191545280, 437925035520, 21158411936256, 1127285473434624, 65884689657464832, 4181915450891501568, 286704379021188538368, 21099339893878107144192, 1659252422924430692327424, 138827012602215571388891136

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144285, A008546 (m=1 column), A144340 (m=3 column).

a(n) = A144285(n+2,2), n>=0.

A144340 Third column (m=3) of triangle S2hat(-4) = A144285.

Original entry on oeis.org

1, 4, 52, 712, 13464, 299520, 8455392, 273091392, 10372138560, 443940231168, 21373226590464, 1136301643620864, 66301616889414144, 4203565310173630464, 287934809159520681984, 21176025476308764622848, 1664420505612432224993280, 139202656243140427345723392

Offset: 0

Wolfdieter Lang Oct 09 2008

A144339 (m=2 column).

a(n)=A144285(n+3,3), n>=0.

A144286 Row sums of triangle A144285 (called S2hat(-4)).

Original entry on oeis.org

1, 5, 41, 561, 10281, 243481, 6965401, 235103417, 9112789817, 399330154617, 19502558109177, 1050527843743993, 61855087724349433, 3952225158444979193, 272326955744655069689, 20128716335079745234425, 1588573078022351774367225, 133323969310853649963402745

Offset: 1

Wolfdieter Lang, Oct 09 2008

Cf. A144285.

a(n) = Sum_{m=1..n} A144285(n,m), n>=1.

A144342 Lower triangular array called S2hat(-5) related to partition number array A144341.

Original entry on oeis.org

1, 5, 1, 55, 5, 1, 935, 80, 5, 1, 21505, 1210, 80, 5, 1, 623645, 29205, 1335, 80, 5, 1, 21827575, 782595, 30580, 1335, 80, 5, 1, 894930575, 27002800, 821095, 31205, 1335, 80, 5, 1, 42061737025, 1058476100, 27963925, 827970, 31205, 1335, 80, 5, 1, 2229272062325, 48782479625

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M32khat(-5)= A144341 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-5). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.

The first three columns are A008543, A144344, A144345.

			[1];[5,1];[55,5,1];[935,80,5,1];[21505,1210,80,5,1];...

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.

Row sums A144343.

A144285 (S2hat(-4)).

a(n,m)=sum(product(|S2(-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. |S2(-5,n,1)|= A013988(n,1) = A008543(n-1) = (6*n-7)(!^6) (6-factorials) for n>=2 and 1 if n=1.

A144284 Partition number array, called M32hat(-4)= 'M32(-4)/M3'= 'A144267/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 4, 1, 36, 4, 1, 504, 36, 16, 4, 1, 9576, 504, 144, 36, 16, 4, 1, 229824, 9576, 2016, 1296, 504, 144, 64, 36, 16, 4, 1, 6664896, 229824, 38304, 18144, 9576, 2016, 1296, 576, 504, 144, 64, 36, 16, 4, 1, 226606464, 6664896, 919296, 344736, 254016, 229824, 38304, 18144

Offset: 1

Wolfdieter Lang Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M32hat(-4;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
If M32hat(-4;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-4):= A144285(n,m).

			a(4,3)= 16 = |S2(-4,2,1)|^2. The relevant partition of 4 is (2^2).

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.

A144279 (M32hat(-3) array). A144341 (M32hat(-5) array)

a(n,k)= product(|S2(-4,j,1)|^e(n,k,j),j=1..n) with |S2(-4,n,1)|= A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

Formally a(n,k)= 'M32(-4)/M3' = 'A144267/A036040' (elementwise division of arrays).

cp's OEIS Frontend

A144339 Second column (m=2) of triangle S2hat(-4) = A144285.

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A144340 Third column (m=3) of triangle S2hat(-4) = A144285.

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A144286 Row sums of triangle A144285 (called S2hat(-4)).

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A144342 Lower triangular array called S2hat(-5) related to partition number array A144341.

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A144284 Partition number array, called M32hat(-4)= 'M32(-4)/M3'= 'A144267/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

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