A008279 -id:A008279 - cp's OEIS frontend

A089517 Array used for numerators of g.f.s for column sequences of array A078741 ((3,3)-Stirling2).

1, 18, 9, 432, 1, 672, 14400, 243, 47520, 648000, 27, 36396, 3790800, 38102400, 1, 9765, 5115888, 354715200, 2844979200, 1107, 2499552, 757646784, 39182330880, 263363788800, 54, 546453, 592216272, 123294623040, 5089348454400

Offset: 3

Wolfdieter Lang, Dec 01 2003

The row length sequence for this array is A004396(n-2)=floor((2*n-3)/3), n>=3: [1,1,2,3,3,4,5,5,6,7,7,8,9,9,10,...].

The g.f. G(m,x) for the m-th column sequence (with leading zeros) of array A078741 is given there. The recurrence is G(m,x) = x*(3*fallfac(m-1,2)*G(m-1,x) + 3*(m-2)*G(m-2,x) + G(m-3,x))/(1-fallfac(m,3)*x), m>=4, with inputs G(1,x)=0=G(2,x) and G(3,x)=x/(1-(3*2*1)*x); where fallfac(n,m) := A008279(n,m) (falling factorials). Computed from the Blasiak et al. reference, eqs. (20) and (21) with r=3: recurrence for S_{3,3}(n,k).

			[1]; [18]; [9,423]; [1,672,14400]; [243,47520,648000]; ...
G(4,x)/(x^2) = 18/((1-3*2*1*x)*(1-4*3*2*x)). kmax(4)=0, hence P(4,x)=a(4,0)=18; x^2 from x^ceiling(4/3).

W. Lang, First 11 rows.

a(n, m) from: sum(a(n, m)*x^m, m=0..kmax(n)) = G(n, x)* product(1-fallfac(p, 3)*x, p=3..n)/x^ceiling(n/3), n>=3, with G(n, x) defined from the recurrence given above and kmax(n) := A004523(n-3)= floor(2*(n-3)/3) = A004396(n-3)-1.

A090217 A generalization of triangle A071951 (Legendre-Stirling).

Original entry on oeis.org

1, 120, 1, 14400, 840, 1, 1728000, 619200, 3360, 1, 207360000, 447552000, 9086400, 10080, 1, 24883200000, 322444800000, 23345280000, 76824000, 25200, 1, 2985984000000, 232185139200000, 59152550400000, 539602560000, 457848000

Offset: 1

Wolfdieter Lang, Dec 01 2003

This is the fourth member of the family A071951 (Legendre-Stirling,(2,2) case), A089504((3,3)-case), A090215 ((4,4)-case).

This triangle underlies the array entry A090216 ((5,5)-generalized Stirling2).

			Triangle starts:
[1];
[120,1];
[14400,840,1];
[1728000,619200,3360,1];
...

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
W. Lang, First 5 rows.

The column sequences (without leading zeros) are powers of 120, etc.

max = 10; f[m_] := 1/Product[1 - FactorialPower[r + 4, 5]*x, {r, 1, m}]; col[m_] := CoefficientList[f[m] + O[x]^(max - m + 1), x]; a[n_, m_] := col[m][[n - m + 1]]; Table[a[n, m], {n, 1, max}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 02 2016 *)

G.f. for m-th column (without leading zeros and m>=1) is 1/product(1-fallfac(r+4, 5)*x, r=1..m) with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m)=sum(A090435(m, p)*fallfac(p, 5)^(n-m), p=1..m)/D(m) if n>=m>=1 else 0; with D(m) := A090436(m).

A122851 Number triangle T(n,k) = C(k,n-k)*(n-k)!.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 6, 4, 1, 0, 0, 0, 6, 12, 5, 1, 0, 0, 0, 0, 24, 20, 6, 1, 0, 0, 0, 0, 24, 60, 30, 7, 1, 0, 0, 0, 0, 0, 120, 120, 42, 8, 1, 0, 0, 0, 0, 0, 120, 360, 210, 56, 9, 1, 0, 0, 0, 0, 0, 0, 720, 840, 336, 72, 10, 1

Offset: 0

Paul Barry, Sep 14 2006

Row sums are A122852.

Triangle T(n,k), read by rows, given by (0,1,-1,0,0,1,-1,0,0,1,-1,0,0,1,...) DELTA (1,0,0,-1,2,0,0,-2,3,0,0,-3,4,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 2, 1;
  0, 0, 2, 3,  1;
  0, 0, 0, 6,  4,  1;
  0, 0, 0, 6, 12,  5, 1;
  0, 0, 0, 0, 24, 20, 6, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A094587, A242653.

T(2n,n) gives A000142.

/* As triangle: */ [[Binomial(k,n-k)*Factorial(n-k): k in [0..n]]: n in [0.. 7]]; // Vincenzo Librandi, Apr 24 2015

Flatten[Table[Binomial[k,n-k](n-k)!,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 16 2012 *)

Number triangle T(n,k) = [k<=n]*k!/(2k-n)!.

T(n,k) = A008279(k,n-k). - Danny Rorabaugh, Apr 23 2015

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.

A145364 Lower triangular array, called S1hat(-2), related to partition number array A145363.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 0, 6, 2, 1, 0, 4, 6, 2, 1, 0, 4, 12, 6, 2, 1, 0, 0, 12, 12, 6, 2, 1, 0, 0, 8, 28, 12, 6, 2, 1, 0, 0, 8, 24, 28, 12, 6, 2, 1, 0, 0, 0, 24, 56, 28, 12, 6, 2, 1, 0, 0, 0, 16, 56, 56, 28, 12, 6, 2, 1, 0, 0, 0, 16, 48, 120, 56, 28, 12, 6, 2, 1, 0, 0, 0, 0, 48, 112, 120, 56, 28, 12, 6, 2, 1

Offset: 1

Wolfdieter Lang, Oct 17 2008

If in the partition array M31hat(-2):=A145363 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-2). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first column is [1,2,2,0,0,0,...]= A008279(2,n-1), n>=1.

			Triangle begins:
  [1];
  [2,1];
  [2,2,1];
  [0,6,2,1];
  [0,4,6,2,1];
  ...

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.

Cf. A145365 (row sums).

a(n,m) = sum(product(S1(-2;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. S1(-2,n,1)= A008279(2,n-1) = [1,2,2,0,0,0,...], n>=1.

A145367 Lower triangular array, called S1hat(-3), related to partition number array A145366.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 6, 15, 3, 1, 0, 24, 15, 3, 1, 0, 54, 51, 15, 3, 1, 0, 36, 108, 51, 15, 3, 1, 0, 36, 198, 189, 51, 15, 3, 1, 0, 0, 360, 360, 189, 51, 15, 3, 1, 0, 0, 324, 846, 603, 189, 51, 15, 3, 1, 0, 0, 216, 1296, 1332, 603, 189, 51, 15, 3, 1, 0, 0, 216, 2484, 2754, 2061, 603, 189

Offset: 1

Wolfdieter Lang, Oct 17 2008

If in the partition array M31hat(-3):=A145366 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-3). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first column is [1,3,6,6,0,0,0,...]= A008279(3,n-1), n>=1.

			Triangle begins:
  [1];
  [3,1];
  [6,3,1];
  [6,15,3,1];
  [0,24,15,3,1];
  ...

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.

Cf. A145368 (row sums).

a(n,m) = sum(product(S1(-3;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. S1(-3,n,1)= A008279(3,n-1) = [1,3,6,6,0,0,0,...], n>=1.

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.

A145370 Lower triangular array, called S1hat(-4), related to partition number array A145369.

Original entry on oeis.org

1, 4, 1, 12, 4, 1, 24, 28, 4, 1, 24, 72, 28, 4, 1, 0, 264, 136, 28, 4, 1, 0, 384, 456, 136, 28, 4, 1, 0, 864, 1344, 712, 136, 28, 4, 1, 0, 576, 4128, 2112, 712, 136, 28, 4, 1, 0, 576, 7488, 7968, 3136, 712, 136, 28, 4, 1, 0, 0, 13248, 20544, 11040, 3136, 712, 136, 28, 4, 1, 0, 0

Offset: 1

Wolfdieter Lang, Oct 17 2008

If in the partition array M31hat(-4):=A145369 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-4). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first column is [1,4,12,24,24,0,0,0,...]= A008279(4,n-1), n>=1.

			Triangle begins:
  [1];
  [4,1];
  [12,4,1];
  [24,28,4,1];
  [24,72,28,4,1];
  ...

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.

Cf. A145371 (row sums).

a(n,m) = sum(product(S1(-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, Y and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S1(-4,n,1)= A008279(4,n-1) = [1,4,12,24,24,0,0,0,...], n>=1.

A145372 Partition number array, called M31hat(-5).

Original entry on oeis.org

1, 5, 1, 20, 5, 1, 60, 20, 25, 5, 1, 120, 60, 100, 20, 25, 5, 1, 120, 120, 300, 400, 60, 100, 125, 20, 25, 5, 1, 0, 120, 600, 1200, 120, 300, 400, 500, 60, 100, 125, 20, 25, 5, 1, 0, 0, 600, 2400, 3600, 120, 600, 1200, 1500, 2000, 120, 300, 400, 500, 625, 60, 100, 125, 20

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 1,2,3,4,5 or 6 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
Fifth member (K=5) 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 A144879 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144879/A036040'. E.g. a(4,3)= 25 = 75/3 = A144879(4,3)/A036040(4,3).
If M31hat(-5;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-5):= A145373.

			Triangle begins;
  [1];
  [5,1];
  [20,5,1];
  [60,20,25,5,1];
  [120,60,100,20,25,5,1];
  ...
a(4,3)= 25 = 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. A145369 (M31hat(-4)).

a(n,k) = product(S1(-5;j,1)^e(n,k,j),j=1..n) with S1(-5;n,1) = A008279(5,n-1) = [1,5,20,60,120,120,0,0,0,...], 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

A089517 Array used for numerators of g.f.s for column sequences of array A078741 ((3,3)-Stirling2).

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A090217 A generalization of triangle A071951 (Legendre-Stirling).

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Mathematica

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A122851 Number triangle T(n,k) = C(k,n-k)*(n-k)!.

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Magma

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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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A145364 Lower triangular array, called S1hat(-2), related to partition number array A145363.

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A145367 Lower triangular array, called S1hat(-3), related to partition number array A145366.

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

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A145370 Lower triangular array, called S1hat(-4), related to partition number array A145369.

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