A144280 -id:A144280 - cp's OEIS frontend

A144282 Second column (m=2) of triangle S2hat(-3) = A144280.

1, 3, 30, 294, 4599, 81081, 1837836, 47609100, 1449052605, 49597258095, 1908702935370, 81045039516210, 3775424041048275, 191061229964098725, 10445176764205306200, 613084928455370309400, 38459158233446064893625, 2567221442902306374982875

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144280, A008545 (m=1 column), A144283 (m-3 column).

a(n) = A144280(n+2,2), n>=0.

A144283 Third column (m=3) of triangle S2hat(-3) = A144280.

Original entry on oeis.org

1, 3, 30, 321, 4788, 84483, 1892835, 48681864, 1471877946, 50200415496, 1925842959810, 81617639962320, 3796524406060155, 191934556035324795, 10484757631267725960, 615052198937234606625, 38564916668583937458750, 2573353724722205851069875

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A144280, A144282 (m=2 column).

a(n) = A144280(n+3,3), n>=0.

A144281 Row sums of triangle A144280 (called S2hat(-3)).

Original entry on oeis.org

1, 4, 25, 265, 3793, 70789, 1600429, 42811078, 1316981794, 45858363502, 1781106761686, 76349861206627, 3579325969675135, 182170488961796899, 10002072131444031160, 589261706147357617009, 37076870479319887575253, 2481442922857293505594237

Offset: 1

Wolfdieter Lang, Oct 09 2008

Cf. A144280.

a(n) = Sum_{m=1..n} A144280(n,m), n>=1.

A144285 Lower triangular array called S2hat(-4) related to partition number array A144284.

Original entry on oeis.org

1, 4, 1, 36, 4, 1, 504, 52, 4, 1, 9576, 648, 52, 4, 1, 229824, 12888, 712, 52, 4, 1, 6664896, 286272, 13464, 712, 52, 4, 1, 226606464, 8182944, 299520, 13720, 712, 52, 4, 1, 8837652096, 266366016, 8455392, 301824, 13720, 712, 52, 4, 1, 388856692224, 10191545280, 273091392

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M32khat(-4)= A144284 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-4). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.

The first three columns are A008546, A144339, A144340.

			[1];[4,1];[36,4,1];[504,52,4,1];[9576,648,52,4,1];...

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.

Row sums A144286.

A144280 (S2hat(-3)), A144342 (S2hat(-5)).

a(n,m)=sum(product(|S2(-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. |S2(-4,n,1)|= A011801(n,1) = A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if n=1.

A144279 Partition number array, called M32hat(-3)= 'M32(-3)/M3'= 'A143173/A036040', related to A000369(n,m)= |S2(-3;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 3, 1, 21, 3, 1, 231, 21, 9, 3, 1, 3465, 231, 63, 21, 9, 3, 1, 65835, 3465, 693, 441, 231, 63, 27, 21, 9, 3, 1, 1514205, 65835, 10395, 4851, 3465, 693, 441, 189, 231, 63, 27, 21, 9, 3, 1, 40883535, 1514205, 197505, 72765, 53361, 65835, 10395, 4851, 2079, 1323, 3465

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(-3;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(-3;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-3):= A144280(n,m).

			a(4,3) = 9 = |S2(-3,2,1)|^2. The relevant partition of 4 is (2^2).

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. A036040, A143173, A134278, A000041, A008545.

Cf. A144274 (M32hat(-2) array), A144284 (M32hat(-4) array).

a(n,k) = Product_{j=1..n} |S2(-3,j,1)|^e(n,k,j), with |S2(-3,n,1)|= A008545(n-1) = (4*n-5)(!^4) (4-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(-3)/M3' = 'A143173/A036040' (elementwise division of arrays).

cp's OEIS Frontend

A144282 Second column (m=2) of triangle S2hat(-3) = A144280.

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A144283 Third column (m=3) of triangle S2hat(-3) = A144280.

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A144281 Row sums of triangle A144280 (called S2hat(-3)).

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A144285 Lower triangular array called S2hat(-4) related to partition number array A144284.

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A144279 Partition number array, called M32hat(-3)= 'M32(-3)/M3'= 'A143173/A036040', related to A000369(n,m)= |S2(-3;n,m)| (generalized Stirling triangle).

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