A134278 -id:A134278 - cp's OEIS frontend

A144268 Partition number array, called M32(-5), related to A013988(n,m)= |S2(-5;n,m)| ( generalized Stirling triangle).

1, 5, 1, 55, 15, 1, 935, 220, 75, 30, 1, 21505, 4675, 2750, 550, 375, 50, 1, 623645, 129030, 70125, 30250, 14025, 16500, 1875, 1100, 1125, 75, 1, 21827575, 4365515, 2258025, 1799875, 451605, 490875, 211750, 144375, 32725, 57750, 13125, 1925, 2625, 105, 1, 894930575

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)=:M32(-5;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, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+4)-ary trees if the outdegree is r >= 0.
If M32(-5;n,k) is summed over those k with fixed number of parts m one obtains triangle A013988(n,m)= |S2(-5;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.

			a(4,3)=75. The relevant partition of 4 is (2^2). The 75 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are 5-ary because r=1 vertices are 5-ary and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 5^2=25 versions due to the two 5-ary root vertices.

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.

Cf. A144267 (M32(-4) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-5,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-5,j,1)|^e(n,k,j),j=1..n), with |S2(-5,n,1)|= A008543(n-1) = (6*n-7)(!^6) (6-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. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).

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).

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).

A144354 Partition number array, called M31(4), related to A049352(n,m)= |S1(4;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 4, 1, 20, 12, 1, 120, 80, 48, 24, 1, 840, 600, 800, 200, 240, 40, 1, 6720, 5040, 7200, 4000, 1800, 4800, 960, 400, 720, 60, 1, 60480, 47040, 70560, 84000, 17640, 50400, 28000, 33600, 4200, 16800, 6720, 700, 1680, 84, 1, 604800, 483840, 752640, 940800, 504000, 188160

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 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) =: M31(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,...].
Fourth member (K=4) in the family M31(K) of partition number arrays.
If M31(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(4)|:= A049352.

			[1];[4,1];[20,12,1];[120,80,48,24,1];[840,600,800,200,240,40,1];...
a(4,3)= 48 = 3*|S1(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.

A049377 (row sums).

A144353 (M31(3) array), A144355 (M31(5) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(4;j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)|= A001715(n+2) = (n+2)!/3!, 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. M3(n,k)=A036040.

A144355 Partition number array, called M31(5), related to A049353(n,m)= |S1(5;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 30, 15, 1, 210, 120, 75, 30, 1, 1680, 1050, 1500, 300, 375, 50, 1, 15120, 10080, 15750, 9000, 3150, 9000, 1875, 600, 1125, 75, 1, 151200, 105840, 176400, 220500, 35280, 110250, 63000, 78750, 7350, 31500, 13125, 1050, 2625, 105, 1, 1663200, 1209600, 2116800

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 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) =: M31(5;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,...].
Fifth member (K=5) in the family M31(K) of partition number arrays.
If M31(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(5)|:= A049353.

			[1];[5,1];[30,15,1];[210,120,75,30,1];[1680,1050,1500,300,375,50,1];...
a(4,3)= 75 = 3*|S1(5;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.

A049378 (row sums).

A144354 (M31(4) array), A144356 (M31(6) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(5;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)|= A001720(n+3) = (n+3)!/4!, 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. M3(n,k)=A036040.

A144357 Partition number array, called M31(-1), related to A049403(n,m) = S1(-1;n,m) (generalized Stirling triangle).

Original entry on oeis.org

1, 1, 1, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 0, 15, 10, 1, 0, 0, 0, 0, 0, 0, 15, 0, 45, 15, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 105, 0, 105, 21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 105, 0, 0, 420, 0, 210, 28, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 945, 0, 0, 1260, 0, 378, 36

Offset: 1

Wolfdieter Lang, Oct 09 2008, Oct 28 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) =: M31(-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, ...].
First member (K=1) in the family M31(-K) of partition number arrays.
If M31(-1;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-1) := A049403.

			[1]; [1,1]; [0,3,1]; [0,0,3,6,1]; [0,0,0,0,15,10,1]; ...
a(4,3) = 3 = 3*S1(-1;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. A000085 (row sums).

Cf. A144358 (M31(-2) array).

a(n,k) = (n!/(Product_{j=1..n} e(n,k,j)!*j!^e(n,k,j))*Product_{j=1..n} S1(-1;j,1)^e(n,k,j) = M3(n,k)*Product_{j=1..n} S1(-1;j,1)^e(n,k,j) with S1(-1;n,1) |= A008279(1,n-1) = [1,1,0,...], 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. M3(n,k) = A036040.

A144879 Partition number array, called M31(-5), related to A049411(n,m) = S1(-5;n,m) (generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 20, 15, 1, 60, 80, 75, 30, 1, 120, 300, 1000, 200, 375, 50, 1, 120, 720, 4500, 4000, 900, 6000, 1875, 400, 1125, 75, 1, 0, 840, 12600, 42000, 2520, 31500, 28000, 52500, 2100, 21000, 13125, 700, 2625, 105, 1, 0, 0, 16800, 134400, 126000, 3360, 100800, 336000

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 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) =: M31(-5;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, ...].
First member (K=5) in the family M31(-K) of partition number arrays.
If M31(-5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-5) := A049411.

			[1]; [5,1]; [20,15,1]; [60,80,75,30,1]; [120,300,1000,200,375,50,1]; ...
a(4,3) = 75 = 3*S1(-5;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.

Cf. A049428 (row sums).

Cf. A144878 (M31(-4) array).

a(n,k)=(n!/Product_{j=1..n} (e(n,k,j)!*j!^e(n,k,j))) * Product_{j=1..n} S1(-5;j,1)^e(n,k,j) = M3(n,k) * Product_{j=1..n} S1(-5;j,1)^e(n,k,j), with S1(-5;n,1) = A008279(5,n-1)= [1,5,20,60,120,120,0,...], 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. M3(n,k)=A036040.

A144880 Partition number array, called M31hat(3).

Original entry on oeis.org

1, 3, 1, 12, 3, 1, 60, 12, 9, 3, 1, 360, 60, 36, 12, 9, 3, 1, 2520, 360, 180, 144, 60, 36, 27, 12, 9, 3, 1, 20160, 2520, 1080, 720, 360, 180, 144, 108, 60, 36, 27, 12, 9, 3, 1, 181440, 20160, 7560, 4320, 3600, 2520, 1080, 720, 540, 432, 360, 180, 144, 108, 81, 60, 36, 27

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) =: M31hat(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,...].
This is the third (K=3) member of a family of partition number arrays: A107106, A134133,...

			[1];[3,1];[12,3,1];[60,12,9,3,1];[360,60,36,12,9,3,1];...
a(4,3)= 9 = |S1(3;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.

A144882 (row sums).

A134133 (M31hat(2) array). A144885 (M31hat(4) array).

a(n,k)= product(|S1(3;j,1)|^e(n,k,j),j=1..n) with |S1(3;n,1)|= A046089(1,n) = [1,3,12,60,...], 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.

A144885 Partition number array, called M31hat(4).

Original entry on oeis.org

1, 4, 1, 20, 4, 1, 120, 20, 16, 4, 1, 840, 120, 80, 20, 16, 4, 1, 6720, 840, 480, 400, 120, 80, 64, 20, 16, 4, 1, 60480, 6720, 3360, 2400, 840, 480, 400, 320, 120, 80, 64, 20, 16, 4, 1, 604800, 60480, 26880, 16800, 14400, 6720, 3360, 2400, 1920, 1600, 840, 480, 400, 320

Offset: 1

Wolfdieter Lang Oct 09 2008, Oct 28 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) =: M31hat(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,...].
Fourth member (K=4) in the family M31hat(K) of partition number arrays.
If M31hat(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1hat(4):= A144886.

			[1];[4,1];[20,4,1];[120,20,16,4,1];[840,120,80,20,16,4,1];...
a(4,3)= 16 = |S1(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.

A144887 (row sums).

A144880 (M31hat(3) array). A144886 (S1hat(4)).

a(n,k) = product(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)| = A049352(n,1) = A001715(n+2) = [1,4,20,120,840,6720,...] = (n+2)!/3!, 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.

A145369 Partition number array, called M31hat(-4).

Original entry on oeis.org

1, 4, 1, 12, 4, 1, 24, 12, 16, 4, 1, 24, 24, 48, 12, 16, 4, 1, 0, 24, 96, 144, 24, 48, 64, 12, 16, 4, 1, 0, 0, 96, 288, 24, 96, 144, 192, 24, 48, 64, 12, 16, 4, 1, 0, 0, 0, 288, 576, 0, 96, 288, 384, 576, 24, 96, 144, 192, 256, 24, 48, 64, 12, 16, 4, 1, 0, 0, 0, 0, 576, 0, 0, 288, 576, 384

Offset: 1

Wolfdieter Lang, Oct 17 2008

If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2,3,4 or 5 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
Fourth member (K=4) in the family M31hat(-K) of partition number arrays.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
This array is array A144878 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144878/A036040'. E.g. a(4,3)= 16 = 48/3 = A144878(4,3)/A036040(4,3).
If M31hat(-4;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-4):= A145370.

			Triangle begins:
  [1];
  [4,1];
  [12,4,1];
  [24,12,16,4,1];
  [24,24,48,12,16,4,1];
  ...
a(4,3)= 16 = S1(-4;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. A145366 (M31hat(-3)), A145372 (M31hat(-5)).

a(n,k) = product(S1(-4;j,1)^e(n,k,j),j=1..n) with S1(-4;n,1) = A008279(4,n-1) = [1,4,12,24,24,0,0,0,...], 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.

cp's OEIS Frontend

A144268 Partition number array, called M32(-5), related to A013988(n,m)= |S2(-5;n,m)| ( generalized Stirling triangle).

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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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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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A144354 Partition number array, called M31(4), related to A049352(n,m)= |S1(4;n,m)| (generalized Stirling triangle).

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A144355 Partition number array, called M31(5), related to A049353(n,m)= |S1(5;n,m)| (generalized Stirling triangle).

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A144357 Partition number array, called M31(-1), related to A049403(n,m) = S1(-1;n,m) (generalized Stirling triangle).

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A144879 Partition number array, called M31(-5), related to A049411(n,m) = S1(-5;n,m) (generalized Stirling triangle).

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A144880 Partition number array, called M31hat(3).

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A144885 Partition number array, called M31hat(4).

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