A134278 -id:A134278 - cp's OEIS frontend

A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

1, 1, 6, 1, 18, 66, 1, 144, 264, 1056, 1, 600, 4620, 5280, 22176, 1, 4950, 68640, 110880, 133056, 576576, 1, 26586, 639870, 3141600, 3259872, 4036032, 17873856, 1, 234528, 10759056, 69263040, 105557760, 113008896, 142990848

Offset: 1

Peter Luschny, Mar 09 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = -6,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A134278.
Same partition product with length statistic is A049385.
Diagonal a(A000217) = A008548.
Row sum is A049412.

Peter Luschny, Counting with Partitions.
Peter Luschny, Generalized Stirling_2 Triangles.

Cf. A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404, A157405

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(-5*j - 1).

Offset corrected by Peter Luschny, Mar 14 2009

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

Original entry on oeis.org

1, 3, 1, 21, 9, 1, 231, 84, 27, 18, 1, 3465, 1155, 630, 210, 135, 30, 1, 65835, 20790, 10395, 4410, 3465, 3780, 405, 420, 405, 45, 1, 1514205, 460845, 218295, 169785, 72765, 72765, 30870, 19845, 8085, 13230, 2835, 735, 945, 63, 1, 40883535, 12113640, 5530140, 4074840

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(-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, ...].
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+2)-ary trees if the outdegree is r >= 0.
If M32(-3;n,k) is summed over those k with fixed number of parts m one obtains triangle A000369(n,m) = |S2(-3;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)=27. The relevant partition of 4 is (2^2). The 12 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 ternary because r=1 vertices are ternary (3-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4 versions due to the two ternary 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. A143172 (M32(-2) array), A144267 (M32(-4) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-3,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-3,j,1)|^e(n,k,j),j=1..n), 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. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).

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

Original entry on oeis.org

1, 4, 1, 36, 12, 1, 504, 144, 48, 24, 1, 9576, 2520, 1440, 360, 240, 40, 1, 229824, 57456, 30240, 12960, 7560, 8640, 960, 720, 720, 60, 1, 6664896, 1608768, 804384, 635040, 201096, 211680, 90720, 60480, 17640, 30240, 6720, 1260, 1680, 84, 1, 226606464, 53319168

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(-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, ...].
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+3)-ary trees if the outdegree is r >= 0.
If M32(-4;n,k) is summed over those k with fixed number of parts m one obtains triangle A011801(n,m)= |S2(-4;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)=48. The relevant partition of 4 is (2^2). The 48 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 quaternary because r=1 vertices are quaternary (4-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4^2=16 versions due to the two quaternary 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. A143173 (M32(-3) array), A144268 (M32(-5) array).

a(n,k) = (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-4,j,1)|^e(n,k,j),j=1..n) = M3(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. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).

A144358 Partition number array, called M31(-2), related to A049404(n,m) = S1(-2;n,m) (generalized Stirling triangle).

Original entry on oeis.org

1, 2, 1, 2, 6, 1, 0, 8, 12, 12, 1, 0, 0, 40, 20, 60, 20, 1, 0, 0, 0, 40, 0, 240, 120, 40, 180, 30, 1, 0, 0, 0, 0, 0, 0, 280, 840, 0, 840, 840, 70, 420, 42, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2240, 0, 0, 1120, 6720, 1680, 0, 2240, 3360, 112, 840, 56, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2240, 0, 0

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(-2;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=2) in the family M31(-K) of partition number arrays.
If M31(-2;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-2) := A049404.

			[1]; [2,1]; [2,6,1]; [0,8,12,12,1]; [0,0,40,20,60,20,1]; ...
a(4,3) = 12 = 3*S1(-2;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. A049425 (row sums).

Cf. A144357 (M31(-1) array), A144877 (M31(-3) array).

a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(-2;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-2;j,1)^e(n,k,j),j=1..n) with S1(-2;n,1)|= A008279(2,n-1)= [1,2,2,0,...], 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.

A144877 Partition number array, called M31(-3), related to A049410(n,m) = S1(-3;n,m) (generalized Stirling triangle).

Original entry on oeis.org

1, 3, 1, 6, 9, 1, 6, 24, 27, 18, 1, 0, 30, 180, 60, 135, 30, 1, 0, 0, 270, 360, 90, 1080, 405, 120, 405, 45, 1, 0, 0, 0, 1260, 0, 1890, 2520, 5670, 210, 3780, 2835, 210, 945, 63, 1, 0, 0, 0, 0, 1260, 0, 0, 10080, 11340, 30240, 0, 7560, 10080, 45360, 8505, 420, 10080, 11340

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(-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, ...].
First member (K=3) in the family M31(-K) of partition number arrays.
If M31(-3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-3) := A049410.

			[1]; [3,1]; [6,9,1]; [6,24,27,18,1]; [0,30,180,60,135,30,1]; ...
a(4,3) = 27 = 3*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.

Cf. A049426 (row sums).

Cf. A144358 (M31(-2) array), A144878 (M31(-4) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(-3;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-3;j,1)^e(n,k,j),j=1..n) with S1(-3;n,1)|= A008279(3,n-1)= [1,3,6,6,0,...], 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.

A144878 Partition number array, called M31(-4), related to A049424(n,m) = S1(-4;n,m) (generalized Stirling triangle).

Original entry on oeis.org

1, 4, 1, 12, 12, 1, 24, 48, 48, 24, 1, 24, 120, 480, 120, 240, 40, 1, 0, 144, 1440, 1440, 360, 2880, 960, 240, 720, 60, 1, 0, 0, 2016, 10080, 504, 10080, 10080, 20160, 840, 10080, 6720, 420, 1680, 84, 1, 0, 0, 0, 16128, 20160, 0, 16128, 80640, 80640, 161280, 1344, 40320

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, ...].
First 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) := A049424.

			[1]; [4,1]; [12,12,1]; [24,48,48,24,1]; [24,120,480,120,240,40,1]; ...
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.

Cf. A049427 (row sums).

Cf. A144877 (M31(-3) array), A144879 (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)= A008279(4,n-1)= [1,4,12,12,24,24,0,...], 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.

A145363 Partition number array, called M31hat(-2).

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 0, 2, 4, 2, 1, 0, 0, 4, 2, 4, 2, 1, 0, 0, 0, 4, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 4, 8, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 4, 8, 16, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 8, 16, 0, 0, 4, 8, 16, 0, 4, 8, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0

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 or 3 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
Second member (K=2) 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 A144358 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144358/A036040'. E.g. a(4,3)= 4 = 12/3 = A144358(4,3)/A036040(4,3).
If M31hat(-2;n,k) is summed over those k belonging to partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-2):= A145364.

			Triangle begins
  [1];
  [2,1];
  [2,2,1];
  [0,2,4,2,1];
  [0,0,4,2,4,2,1];
  ...
a(4,3)= 4 = S1(-2;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. A145361 (M31hat(-1)). A145366 (M31hat(-3)).

a(n,k) = product(S1(-2;j,1)^e(n,k,j),j=1..n) with S1(-2;n,1) = A008279(2,n-1) = [1,2,1,0,0,0,...], 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.

A145366 Partition number array, called M31hat(-3).

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 6, 6, 9, 3, 1, 0, 6, 18, 6, 9, 3, 1, 0, 0, 18, 36, 6, 18, 27, 6, 9, 3, 1, 0, 0, 0, 36, 0, 18, 36, 54, 6, 18, 27, 6, 9, 3, 1, 0, 0, 0, 0, 36, 0, 0, 36, 54, 108, 0, 18, 36, 54, 81, 6, 18, 27, 6, 9, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 108, 216, 0, 0, 36, 54, 108, 162, 0, 18, 36, 54, 81

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 or 4 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
Third member (K=3) 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 A144877 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144877/A036040'. E.g. a(4,3)= 9 = 27/3 = A144877(4,3)/A036040(4,3).
If M31hat(-3;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-3):= A145367.

			Triangle begins:
  [1];
  [3,1];
  [6,3,1];
  [6,6,9,3,1];
  [0,6,18,6,9,3,1];
  ...
a(4,3)= 9 = S1(-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. A145363 (M31hat(-2)). A145369 (M31hat(-4)).

a(n,k) = product(S1(-3;j,1)^e(n,k,j),j=1..n) with S1(-3;n,1) = A008279(3,n-1) = [1,3,6,6,0,0,0,...], 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.

A143171 Partition number array, called M32(-1), related to A001497(n-1,m-1) = |S2(-1;n,m)| (generalized Stirling2 triangle).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 15, 12, 3, 6, 1, 105, 75, 30, 30, 15, 10, 1, 945, 630, 225, 90, 225, 180, 15, 60, 45, 15, 1, 10395, 6615, 2205, 1575, 2205, 1575, 630, 315, 525, 630, 105, 105, 105, 21, 1, 135135, 83160, 26460, 17640, 7875, 26460, 17640, 12600, 3150, 2520, 5880, 6300

Offset: 1

Wolfdieter Lang, Oct 09 2008, Dec 04 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(-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, ...].
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_{j=1..n} e(n,k,j). The special (enk)-forest is composed of m rooted increasing r-ary trees if the outdegree is r >= 0.
This generalizes the array of multinomials called M_3 in Abramowitz-Stegun, pp. 831-2. M_3 = A036040.
If M32(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle A001497(n-1,m-1) = |S2(-1;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) = 3. The relevant partition of 4 is (2^2). The 3 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing unary trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are unary because r=1 vertices are unary (1-ary) and for the leaves (r=0) the arity does not matter.

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. A143173 M32(-2) array.

a(n,k) = (n!/Product_{j=1..n} e(n,k,j)!*j!^e(n,k,j)) * Product_{j=1..n} |S2(-1,j,1)|^e(n,k,j) = M3(n,k)*Product_{j=1..n} |S2(-1,j,1)|^e(n,k,j), 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. Exponents 0 can be omitted due to 0!=1. M3(n,k) := A036040(n,k), k=1..p(n), p(n) := A000041(n).

A143172 Partition number array, called M32(-2), related to A004747(n,m) = |S2(-2;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 2, 1, 10, 6, 1, 80, 40, 12, 12, 1, 880, 400, 200, 100, 60, 20, 1, 12320, 5280, 2400, 1000, 1200, 1200, 120, 200, 180, 30, 1, 209440, 86240, 36960, 28000, 18480, 16800, 7000, 4200, 2800, 4200, 840, 350, 420, 42, 1, 4188800, 1675520, 689920, 492800, 224000, 344960

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(-2;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+1)-ary trees if the outdegree is r>=0.
If M32(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle A004747(n,m)= |S2(-2;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)=12. The relevant partition of 4 is (2^2). The 12 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 binary because r=1 vertices are binary (2-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4 versions due to the two binary 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. A143171 (M32(-1) array), A143173 (M32(-3) array).

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-2,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-2,j,1)|^e(n,k,j),j=1..n), with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-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).

cp's OEIS Frontend

A157396 A partition product of Stirling_2 type [parameter k = -6] with biggest-part statistic (triangle read by rows).

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

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

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A144358 Partition number array, called M31(-2), related to A049404(n,m) = S1(-2;n,m) (generalized Stirling triangle).

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A144877 Partition number array, called M31(-3), related to A049410(n,m) = S1(-3;n,m) (generalized Stirling triangle).

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A144878 Partition number array, called M31(-4), related to A049424(n,m) = S1(-4;n,m) (generalized Stirling triangle).

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A145363 Partition number array, called M31hat(-2).

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A145366 Partition number array, called M31hat(-3).

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A143171 Partition number array, called M32(-1), related to A001497(n-1,m-1) = |S2(-1;n,m)| (generalized Stirling2 triangle).

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