A004747 -id:A004747 - cp's OEIS frontend

A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0).

1, 5, 1, 55, 15, 1, 935, 295, 30, 1, 21505, 7425, 925, 50, 1, 623645, 229405, 32400, 2225, 75, 1, 21827575, 8423415, 1298605, 103600, 4550, 105, 1, 894930575, 358764175, 59069010, 5235405, 271950, 8330, 140, 1, 42061737025, 17398082625, 3016869625, 289426830, 16929255, 621810, 14070, 180, 1

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A049224; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n, m) = S2p(-5; n,m), a member of a sequence of triangles including S2p(-1; n,m) = A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) = A008277(n,m) (Stirling 2nd kind). T(n, 1) = A008543(n-1).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			Triangle begins as:
          1;
          5,         1;
         55,        15,        1;
        935,       295,       30,       1;
      21505,      7425,      925,      50,      1;
     623645,    229405,    32400,    2225,     75,     1;
   21827575,   8423415,  1298605,  103600,   4550,   105,    1;
  894930575, 358764175, 59069010, 5235405, 271950,  8330,  140,   1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Peter Luschny, The Bell transform
Index entries for sequences related to Bessel functions or polynomials

Cf. A008277, A008543, A049224, A264428.
Cf. A028844 (row sums).
Triangles with the recurrence T(n,k) = (m*(n-1)-k)*T(n-1,k) + T(n-1,k-1): A010054 (m=1), A001497 (m=2), A004747 (m=3), A000369 (m=4), A011801 (m=5), this sequence (m=6).

function T(n,k) // T = A013988
  if k eq 0 then return 0;
  elif k eq n then return 1;
  else return (6*(n-1)-k)*T(n-1,k) + T(n-1,k-1);
  end if;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2023

(* First program *)
rows = 10;
b[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
A = Table[b[n, m], {n, 1, rows}, {m, 1, rows}] // Inverse // Abs;
A013988 = Table[A[[n, m]], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==0, 0, If[k==n, 1, (6*(n-1) -k)*T[n-1,k] +T[n-1, k-1]]];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 03 2023 *)

# uses[inverse_bell_matrix from A264428]
# Adds 1,0,0,0, ... as column 0 at the left side of the triangle.
inverse_bell_matrix(lambda n: factorial(n)*binomial(5, n), 8) # Peter Luschny, Jan 16 2016

T(n, m) = n!*A049224(n, m)/(m!*6^(n-m));

T(n+1, m) = (6*n-m)*T(n, m) + T(n, m-1), for n >= m >= 1, with T(n, m) = 0, n

E.g.f. of m-th column: ((1 - (1-6*x)^(1/6))^m)/m!.

Sum_{k=1..n} T(n, k) = A028844(n).

New name from Peter Luschny, Jan 16 2016

A157402 A partition product of Stirling_2 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 2, 1, 6, 10, 1, 24, 40, 80, 1, 80, 300, 400, 880, 1, 330, 2400, 3600, 5280, 12320, 1, 1302, 15750, 47600, 55440, 86240, 209440, 1, 5936, 129360, 588000, 837760, 1034880, 1675520, 4188800, 1, 26784, 1146040, 5856480

Offset: 1

Peter Luschny, Mar 09 2009, Mar 14 2009

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 2,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143172.
Same partition product with length statistic is A004747.
Diagonal a(A000217) = A008544.
Row sum is A015735.

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

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, 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}(3*j - 1).

A049404 Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0).

Original entry on oeis.org

1, 2, 1, 2, 6, 1, 0, 20, 12, 1, 0, 40, 80, 20, 1, 0, 40, 360, 220, 30, 1, 0, 0, 1120, 1680, 490, 42, 1, 0, 0, 2240, 9520, 5600, 952, 56, 1, 0, 0, 2240, 40320, 48720, 15120, 1680, 72, 1, 0, 0, 0, 123200, 332640, 184800, 35280, 2760, 90, 1, 0, 0, 0, 246400, 1786400

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to triangle A049324.
a(n,1) = A008279(2,n-1). a(n,m) =: S1(-2; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers).
a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A004747(n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			E.g. row polynomial E(3,x) = 2*x+6*x^2+x^3.
Triangle starts:
{1}
{2,  1}
{2,  6,  1}
{0, 20, 12, 1}

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows of the array and more. [From _Wolfdieter Lang_, Oct 17 2008]
Peter Luschny, The Bell transform

Row sums give A049425.

Cf. A004747, A049324.

rows = 11;
a[n_, m_] := BellY[n, m, Table[k! Binomial[2, k], {k, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[bell_matrix from A264428]
# Adds 1,0,0,0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: factorial(n)*binomial(2, n), 8) # Peter Luschny, Jan 16 2016

a(n, m) = n!*A049324(n, m)/(m!*3^(n-m));
a(n, m) = (3*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
a(n, m) = 0, nE.g.f. for m-th column: ((x+x^2+(x^3)/3)^m)/m!.
a(n,m) = n!/(3^m * m!)*(Sum_{i=0..floor(m-n/3)} (-1)^i * binomial(m,i) * binomial(3*m-3*i,n)), 0 for empty sums. - Werner Schulte, Feb 20 2020

New name from Peter Luschny, Jan 16 2016

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.

A203412 Triangle read by rows, a(n,k), n>=k>=1, which represent the s=3, h=1 case of a two-parameter generalization of Stirling numbers arising in conjunction with normal ordering.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 28, 19, 6, 1, 280, 180, 55, 10, 1, 3640, 2260, 675, 125, 15, 1, 58240, 35280, 10360, 1925, 245, 21, 1, 1106560, 658000, 190680, 35385, 4620, 434, 28, 1, 24344320, 14266560, 4090240, 756840, 100065, 9828, 714, 36, 1

Offset: 1

Mark Shattuck, Jan 01 2012

Also the Bell transform of the triple factorial numbers A007559 which adds a first column (1,0,0 ...) on the left side of the triangle. For the definition of the Bell transform see A264428. See A051141 for the triple factorial numbers A032031 and A004747 for the triple factorial numbers A008544 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 23 2015

			Triangle starts:
[    1]
[    1,     1]
[    4,     3,     1]
[   28,    19,     6,    1]
[  280,   180,    55,   10,   1]
[ 3640,  2260,   675,  125,  15,  1]
[58240, 35280, 10360, 1925, 245, 21, 1]

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
T. Mansour, M. Schork, and M. Shattuck, On a new family of generalized Stirling and Bell numbers, Electron. J. Combin. 18 (2011) #P77 (33 pp.).
Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771.

Cf. A007559, A032031, A039683, A051141, A132062, A264428.

A203412 := (n,k) -> (n!*3^n)/(k!*2^k)*add((-1)^j*binomial(k,j)*binomial(n-2*j/3-1, n), j=0..k): seq(seq(A203412(n,k),k=1..n),n=1..9); # Peter Luschny, Dec 21 2015

Table[(n! 3^n)/(k! 2^k) Sum[ (-1)^j Binomial[k, j] Binomial[n - 2 j/3 - 1, n], {j, 0, k}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Dec 23 2015 *)

# uses[bell_transform from A264428]
triplefactorial = lambda n: prod(3*k + 1 for k in (0..n-1))
def A203412_row(n):
    trifact = [triplefactorial(k) for k in (0..n)]
    return bell_transform(n, trifact)
[A203412_row(n) for n in (0..8)] # Peter Luschny, Dec 21 2015

(1) Is given by the recurrence relation
  a(n+1,k) = a(n,k-1)+(3*n-2*k)*a(n,k) if n>=0 and k>=1, along with the initial values a(n,0) = delta_{n,0} and a(0,k) = delta_{0,k} for all n,k>=0.
(2) Is given explicitly by
  a(n,k) = (n!*3^n)/(k!*2^k)*Sum{j=0..k} (-1)^j*C(k,j)*C(n-2*j/3-1,n) for all n>=k>=1.
a(n,1) = A007559(n-1). - Peter Luschny, Dec 21 2015

A265606 Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 45, 23, 6, 1, 0, 585, 275, 65, 10, 1, 0, 9945, 4435, 990, 145, 15, 1, 0, 208845, 89775, 19285, 2730, 280, 21, 1, 0, 5221125, 2183895, 456190, 62965, 6370, 490, 28, 1, 0, 151412625, 62002395, 12676265, 1715490, 171255, 13230, 798, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[1],
[0, 1],
[0, 1, 1],
[0, 5, 3, 1],
[0, 45, 23, 6, 1],
[0, 585, 275, 65, 10, 1],
[0, 9945, 4435, 990, 145, 15, 1],
[0, 208845, 89775, 19285, 2730, 280, 21, 1],

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Peter Luschny, The Bell transform

Cf. A007696, A264428, A264429.

Bell transforms of other multifactorials are: A000369, A004747, A039683, A051141, A051142, A119274, A132062, A132393, A203412.

(* The function BellMatrix is defined in A264428. *)
rows = 10;
M = BellMatrix[Pochhammer[1/4, #] 4^# &, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 23 2019 *)

# uses[bell_transform from A264428]
def A265606_row(n):
    multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
    mfact = [multifact_4_1(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A265606_row(n) for n in (0..7)]

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A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0).

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A157402 A partition product of Stirling_2 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

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A049404 Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0).

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

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A203412 Triangle read by rows, a(n,k), n>=k>=1, which represent the s=3, h=1 case of a two-parameter generalization of Stirling numbers arising in conjunction with normal ordering.

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A265606 Triangle read by rows: The Bell transform of the quartic factorial numbers (A007696).

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