A144270 -id:A144270 - cp's OEIS frontend

A144271 Row sums of triangle A144270 (called S2hat(-1)).

1, 2, 5, 21, 129, 1099, 11647, 148292, 2190302, 36842892, 694276152, 14488348603, 331537914373, 8254117606799, 222087256536602, 6421589319659283, 198565371839434761, 6538245667687424041, 228396692050813678534, 8436237099236470594589

Offset: 1

Wolfdieter Lang, Oct 09 2008

Cf. A144270.

a(n) = Sum_{m=1..n} A144270(n,m), n>=1.

A144272 Second column (m=2) of triangle S2hat(-1) = A144270.

Original entry on oeis.org

1, 1, 4, 18, 129, 1095, 11880, 149940, 2218545, 37147005, 699281100, 14561835750, 332913128625, 8280095098275, 222659485448400, 6434896997529000, 198909798356894625, 6547761582390653625, 228681849947367316500, 8445341134525197152250, 328822530412023653630625

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144270.

Cf. A001147 (column m=1), A144273 (column m=3).

a(n) = A144270(n+2,2), n>=0.

A144273 Third column (m=3) of triangle S2hat(-1) = A144270.

Original entry on oeis.org

1, 1, 4, 19, 132, 1119, 12057, 151560, 2234970, 37355130, 702161460, 14608747440, 333760916475, 8297318063475, 223043540630400, 6444290305953675, 199158606533605950, 6554872218153191625, 228899544451085017125, 8452452568601816775000, 329069272855540304587500

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144270, A144272 (column m=2).

a(n) = A144270(n+3,3), n>=0.

A144275 Lower triangular array called S2hat(-2) related to partition number array A144274.

Original entry on oeis.org

1, 2, 1, 10, 2, 1, 80, 14, 2, 1, 880, 100, 14, 2, 1, 12320, 1140, 108, 14, 2, 1, 209440, 14880, 1180, 108, 14, 2, 1, 4188800, 249280, 15400, 1196, 108, 14, 2, 1, 96342400, 4801280, 255400, 15480, 1196, 108, 14, 2, 1, 2504902400, 108574400, 4888960, 256440, 15512

Offset: 1

Wolfdieter Lang, Oct 09 2008

If in the partition array M32khat(-2)= A144274 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-2). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.

The first three columns are A008544, A144277, A144278.

			Triangle begins:
  [1];
  [2,1];
  [10,2,1];
  [80,14,2,1];
  [880,100,14,2,1];
  ...

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.

Row sums A144276.
A144270 (S2hat(-1)).
Cf. A004747, A008284, A008544.

a(n,m) = Sum_{q=1..p(n,m)} (Product_{j=1..n} |S2(-2;j,1)|^e(n,m,q,j)) 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(-2,n,1)|= A004747(n,1) = A008544(n-1) = (3*n-4)(!^3) (3-factorials) for n>=2 and 1 if n=1.

A144269 Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 15, 3, 1, 1, 1, 105, 15, 3, 3, 1, 1, 1, 945, 105, 15, 9, 15, 3, 1, 3, 1, 1, 1, 10395, 945, 105, 45, 105, 15, 9, 3, 15, 3, 1, 3, 1, 1, 1, 135135, 10395, 945, 315, 225, 945, 105, 45, 15, 9, 105, 15, 9, 3, 1, 15, 3, 1, 3, 1, 1, 1, 2027025, 135135, 10395, 2835

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(-1;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(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-1):= A144270(n,m).

			a(4,3)= 1 = |S2(-1,2,1)|^2. The relevant partition of 4 is (2^2).
[1]; [1,1]; [3,1,1]; [15,3,1,1,1]; [105,15,3,3,1,1,1]; ... [From _Wolfdieter Lang_, Oct 23 2008]

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. A144271 (M32hat(-2) array).

a(n,k)= product(|S2(-1,j,1)|^e(n,k,j),j=1..n) with |S2(-1,n,1)|= A001147(n-1) = (2*n-3)(!^2) (2-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(-1)/M3' = 'A143171/A036040' (elementwise division of arrays).

Corrected all entries. Wolfdieter Lang, Oct 23 2008

cp's OEIS Frontend

A144271 Row sums of triangle A144270 (called S2hat(-1)).

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A144272 Second column (m=2) of triangle S2hat(-1) = A144270.

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A144273 Third column (m=3) of triangle S2hat(-1) = A144270.

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A144275 Lower triangular array called S2hat(-2) related to partition number array A144274.

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A144269 Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle).

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