A062147 -id:A062147 - cp's OEIS frontend

A062137 Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x).

1, 4, -1, 20, -10, 1, 120, -90, 18, -1, 840, -840, 252, -28, 1, 6720, -8400, 3360, -560, 40, -1, 60480, -90720, 45360, -10080, 1080, -54, 1, 604800, -1058400, 635040, -176400, 25200, -1890, 70, -1, 6652800, -13305600, 9313920

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,3,x) = Sum_{m=0..n} a(n,m)*x^m have e.g.f. exp(-z*x/(1-z))/(1-z)^4. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} |A008297(n,m)|*(-x)^m, n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).
These polynomials appear in the radial part of the l=1 (p-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
The unsigned version of this triangle is the triangle of unsigned 2-Lah numbers A143497. - Peter Bala, Aug 25 2008
This matrix (unsigned) is embedded in that for n!*L(n,-3,-x). Introduce 0,0,0 to each unsigned row and then add 1,-2,1,4,2,1 to the beginning of the array as the first three rows to generate n!*L(n,-3,-x). - Tom Copeland, Apr 21 2014
From Wolfdieter Lang, Jul 07 2014: (Start)
The standard Rodrigues formula for the monic generalized Laguerre polynomials (also called Laguerre-Sonin polynomials) is Lm(n,alpha,x) := (-1)^n*n!*L(n,3,x) is x^(-alpha)*exp(x)*(d/dx)^n(exp(-x)*x^(n+alpha)).
Another Rodrigues type formula is Lm(n,alpha,x) = exp(x)*x^(-alpha+n+1)*(-x^2*d/dx)^n*(exp(-x)*x^(alpha+1)). This is derivable from the differential - difference relation of Lm(n,alpha,x) which yields first the creation operator formula -(x*d/dx + (-x + alpha + n + 1))*Lm(n,alpha,x) = Lm(n+1,alpha,x) or in the variable q = log(x) the operator -(d/dq + alpha + n + 1 - exp(q)).
The identity (due to Christoph Mayer) (d/dq - (d/dq)W(q))*f(q) = exp(W(q)*d/dq(exp(-W(q)*f(q)) is then iterated with W(q) = W(alpha,n,q) = exp(q) - (alpha + n + 1)*q and a further change of variables leads then to the given result. (End)

			The triangle a(n,m) begins:
n\m       0        1       2     3    4   5 ...
0:        1
1:        4       -1
2:       20      -10      1
3:      120      -90     18     -1
4:      840     -840    252    -28    1
5:     6720    -8400   3360   -560   40  -1
... Formatted by _Wolfdieter Lang_, Jul 07 2014
For more rows see the link.
n = 2: 2!*L(2,3,x) = 20 - 10*x + x^2.

A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.

Indranil Ghosh, Rows 0..125, flattened
Wolfdieter Lang, First eleven rows of the triangle.
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001715, A061206, A062141-A062144. The row sums (signed) give A062146, the row sums (unsigned) give A062147.
Cf. A143497. - Peter Bala, Aug 25 2008
Cf. A062139, A105278. - Wolfdieter Lang, Jul 07 2014

Flatten[Table[((-1)^m)*n!*Binomial[n+3,n-m]/m!,{n,0,9},{m,0,n}]] (* Indranil Ghosh, Feb 23 2017 *)

row(n) = Vecrev(n!*pollaguerre(n, 3)); \\ Michel Marcus, Feb 06 2021

a(n, m) = ((-1)^m)*n!*binomial(n+3, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^4), m >= 0.

A086885 Lower triangular matrix, read by rows: T(i,j) = number of ways i seats can be occupied by any number k (0<=k<=j<=i) of persons.

Original entry on oeis.org

2, 3, 7, 4, 13, 34, 5, 21, 73, 209, 6, 31, 136, 501, 1546, 7, 43, 229, 1045, 4051, 13327, 8, 57, 358, 1961, 9276, 37633, 130922, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 10, 91, 748, 5509, 36046, 207775, 1047376, 4596553, 17572114, 11, 111, 1021, 8501

Offset: 1

Hugo Pfoertner, Aug 22 2003

Compare with A088699. - Peter Bala, Sep 17 2008

T(m, n) gives the number of matchings in the complete bipartite graph K_{m,n}. - Eric W. Weisstein, Apr 25 2017

			One person:
T(1,1)=a(1)=2: 0,1 (seat empty or occupied);
T(2,1)=a(2)=3: 00,10,01 (both seats empty, left seat occupied, right seat occupied).
Two persons:
T(2,2)=a(3)=7: 00,10,01,20,02,12,21;
T(3,2)=a(5)=13: 000,100,010,001,200,020,002,120,102,012,210,201,021.
Triangle starts:
  2;
  3  7;
  4 13  34;
  5 21  73 209;
  6 31 136 501 1546;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)
Ed Jones, Number of seatings, discussion in newsgroup sci.math, Aug 9, 2003.
R. J. Mathar, The number of binary nxm matrices with at most k 1's in each row or columns, Table 1.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Luca Zecchini, Tobias Bleifuß, Giovanni Simonini, Sonia Bergamaschi, and Felix Naumann, Determining the Largest Overlap between Tables, Proc. ACM Manag. Data (SIGMOD 2024) Vol. 2, No. 1, Art. 48. See p. 48:6.
Index entries for sequences related to Laguerre polynomials

Diagonal: A002720, first subdiagonal: A000262, 2nd subdiagonal: A052852, 3rd subdiagonal: A062147, 4th subdiagonal: A062266, 5th subdiagonal: A062192, 2nd row/column: A002061. With column 0: A176120.

[Factorial(k)*Evaluate(LaguerrePolynomial(k, n-k), -1): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021

A086885 := proc(n,k)
    add( binomial(n,j)*binomial(k,j)*j!,j=0..min(n,k)) ;
end proc: # R. J. Mathar, Dec 19 2014

Table[Table[Sum[k! Binomial[n, k] Binomial[j, k], {k, 0, j}], {j, 1, n}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Jul 09 2015 *)
Table[m! LaguerreL[m, n - m, -1], {n, 10}, {m, n}] // Flatten (* Eric W. Weisstein, Apr 25 2017 *)

T(i, j) = j!*pollaguerre(j, i-j, -1); \\ Michel Marcus, Feb 23 2021

flatten([[factorial(k)*gen_laguerre(k, n-k, -1) for k in [1..n]] for n in (1..10)]) # G. C. Greubel, Feb 23 2021

a(n) = T(i, j) with n=(i*(i-1))/2+j; T(i, 1)=i+1, T(i, j)=T(i, j-1)+i*T(i-1, j-1) for j>1.
The role of seats and persons may be interchanged, so T(i, j)=T(j, i).
T(i, j) = j!*LaguerreL(j, i-j, -1). - Vladeta Jovovic, Aug 25 2003
T(i, j) = Sum_{k=0..j} k!*binomial(i, k)*binomial(j, k). - Vladeta Jovovic, Aug 25 2003

A143497 Triangle of unsigned 2-Lah numbers.

Original entry on oeis.org

1, 4, 1, 20, 10, 1, 120, 90, 18, 1, 840, 840, 252, 28, 1, 6720, 8400, 3360, 560, 40, 1, 60480, 90720, 45360, 10080, 1080, 54, 1, 604800, 1058400, 635040, 176400, 25200, 1890, 70, 1, 6652800, 13305600, 9313920, 3104640, 554400, 55440, 3080, 88, 1

Offset: 2

Peter Bala, Aug 25 2008

For a signed version of this triangle see A062137. The unsigned 2-Lah number L(2; n,k) gives the number of partitions of the set {1, 2, ..., n} into k ordered lists with the restriction that the elements 1 and 2 must belong to different lists. More generally, the unsigned r-Lah number L(r; n, k) gives the number of partitions of the set {1, 2, ..., n} into k ordered lists with the restriction that the elements 1, 2, ..., r belong to different lists. If r = 1 there is no restriction and we obtain the unsigned Lah numbers A105278. For other cases see A143498 (r=3) and A143499 (r=4). We make some remarks on the general case.
The unsigned r-Lah numbers occur as connection constants in the generalized Lah identity (x + 2*r - 1)*(x + 2*r)*...*(x + 2*r + n - r - 2) = Sum_{k=r..n} L(r; n, k)*(x - 1)*(x - 2)*...*(x - k + r) for n >= r and where any empty products are taken equal to 1 (for a bijective proof of the identity, follow the proof of [Petkovsek and Pisanski] but restrict the first r of the Argonauts to different paths).
The unsigned r-Lah numbers satisfy the same recurrence as the unsigned Lah numbers, namely, L(r; n, k) = (n + k - 1)*L(r; n - 1,k) + L(r; n - 1,k - 1), but with the boundary conditions: L(r; n, k) = 0 if n < r or if k < r; L(r; r, r) = 1. The recurrence has the explicit solution L(r; n, k) = ((n - r)!/(k - r)!)*binomial(n + r - 1, k + r - 1) for n, k >= r. It follows that the unsigned r-Lah numbers have 'vertical' generating functions for k >= r of the form Sum_{n>=k} L(r; n, k)*t^n/(n -r)! = 1/(k - r)!*t^k/(1 - t)^(k + r). This yields the e.g.f. for the array of unsigned r-restricted Lah numbers in the form: Sum_{n,k>=r} L(r; n, k)*x^k*t^n/(n-r)! = (x*t)^r * 1/(1 - t)^(2*r) * exp(x*t/(1 - t)) = (x*t)^r (1 + (2*r + x)*t + (2r*(2*r + 1) + 2*(2*r + 1)*x + x^2)*t^2/2! + ...).
The array of unsigned r-Lah numbers begins
  1
  2r                  1
  2r*(2r+1)           2*(2r+1)           1
  2r*(2r+1)*(2r+2)    3*(2r+1)*(2r+2)    3*(2r+2)      1
 ...
and equals exp(D(r)), where D(r) is the array with the sequence (2*r, 2*(2*r + 1), 3*(2*r + 2), 4*(2*r + 3), ...) on the main subdiagonal and zeros everywhere else.
The unsigned r-Lah numbers are related to the r-Stirling numbers: the lower triangular array of unsigned r-Lah numbers may be expressed as the matrix product St1(r) * St2(r), where St1(r) and St2(r) denote the arrays of r-Stirling numbers of the first and second kind respectively. The theory of r-Stirling numbers is developed in [Broder]. See A143491 - A143496 for tables of r-Stirling numbers. An alternative factorization for the array is as St1 * P^(2r - 2) * St2, where P denotes Pascal's triangle, A007318, St1 is the triangle of unsigned Stirling numbers of the first kind, abs(A008275) and St2 denotes the triangle of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]).
The array of unsigned r-Lah numbers is an example of the fundamental matrices sketched in A133314. So redefining the offset as n=0, given matrices A and B with A(n, k) = T(n, k)*a(n - k) and B(n, k) = T(n, k)*b(n - k), then A*B = C where C(n, k) = T(n,k)*[a(.) + b(.)]^(n - k), umbrally. An e.g.f. for the row polynomials of A is exp(x*t) exp{-x*t*[a*t/(a*t - 1)]}/(1 - a*t)^4 = exp(x*t) exp[(.)!*Laguerre(., 3, -x*t)* a(.)*t)], umbrally. - Tom Copeland, Sep 19 2008

			Triangle begins:
=========================================
n\k |     2     3     4     5     6     7
----+------------------------------------
  2 |     1
  3 |     4     1
  4 |    20    10     1
  5 |   120    90    18     1
  6 |   840   840   252    28     1
  7 |  6720  8400  3360   560    40     1
 ...
T(4,3) = 10. The ten partitions of {1,2,3,4} into 3 ordered lists such that the elements 1 and 2 lie in different lists are: {1}{2}{3,4} and {1}{2}{4,3}, {1}{3}{2,4} and {1}{3}{4,2}, {1}{4}{2,3} and {1}{4}{3,2}, {2}{3}{1,4} and {2}{3}{4,1}, {2}{4}{1,3} and {2}{4}{3,1}. The remaining two partitions {3}{4}{1,2} and {3}{4}{2,1} are not allowed because the elements 1 and 2 belong to the same block.

Muniru A Asiru, Table of n, a(n) for n = 2..4951 Rows n = 2..100
A. Z. Broder, The r-Stirling numbers, Report CS-TR-82-949, Stanford University, Department of Computer Science, 1982.
A. Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
Gi-Sang Cheon and Ji-Hwan Jung, r-Whitney numbers of Dowling lattices, Discrete Math., 312 (2012), 2337-2348.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Tech Report TR 99-05, 1999.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).
G. Nyul and G. Rácz, The r-Lah numbers, Discrete Mathematics, 338 (2015), 1660-1666.
Marko Petkovsek and Tomaz Pisanski, Combinatorial interpretation of unsigned Stirling and Lah numbers, University of Ljubljana, Preprint series, Vol. 40 (2002), 837.
Jose L. Ramirez and M. Shattuck, A (p, q)-Analogue of the r-Whitney-Lah Numbers, Journal of Integer Sequences, 19, 2016, #16.5.6.
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.
M. Shattuck, Generalized r-Lah numbers, arXiv:1412.8721 [math.CO], 2014.

Cf. A001715 (column 2), A007318, A008275, A008277, A061206 (column 3), A062137, A062141 - A062144 ( column 4 to column 7), A062146 (alt. row sums), A062147 (row sums), A105278 (unsigned Lah numbers), A143491, A143494, A143498, A143499.

T:=Flat(List([2..10],n->List([2..n],k->(Factorial(n-2)/Factorial(k-2))*Binomial(n+1,k+1)))); # Muniru A Asiru, Nov 27 2018

T := (n, k) -> ((n-2)!/(k-2)!)*binomial(n+1, k+1):
for n from 2 to 11 do seq(T(n, k), k = 2..n) od;

T[n_, k_] := (n-2)!/(k-2)!*Binomial[n+1, k+1]; Table[T[n, k], {n,2,10}, {k,2,n}] // Flatten (* Amiram Eldar, Nov 27 2018 *)

create_list((n - 2)!/(k - 2)!*binomial(n + 1, k + 1), n, 2, 12, k, 2, n); /* Franck Maminirina Ramaharo, Nov 27 2018 */

T(n, k) = ((n - 2)!/(k - 2)!)*C(n+1, k+1), for n, k >= 2.
Recurrence: T(n, k) = (n + k - 1)*T(n-1, k) + T(n-1, k-1) for n, k >= 2, with the boundary conditions: T(n, k) = 0 if n < 2 or k < 2; T(2, 2) = 1.
E.g.f. for column k: Sum_{n>=k} T(n, k)*t^n/(n - 2)! = 1/(k - 2)!*t^k/(1 - t)^(k+2) for k >= 2.
E.g.f: Sum_{n=2..inf} Sum_{k=2..n} T(n, k)*x^k*t^n/(n - 2)! = (x*t)^2/(1 - t)^4* exp(x*t/(1 - t)) = (x*t)^2*(1 + (4 + x)*t + (20 + 10*x + x^2)*t^2/2! + ... ).
Generalized Lah identity: (x + 3)*(x + 4)*...*(x + n) = Sum_{k = 2..n} T(n, k)*(x - 1)*(x - 2)*...*(x - k + 2).
The polynomials 1/n!*Sum_{k=2..n+2} T(n+2, k)*(-x)^(k - 2) for n >= 0 are the generalized Laguerre polynomials Laguerre(n,3,x). See A062137.
Array = A143491 * A143494 = abs(A008275) * (A007318)^2 * A008277 (apply Theorem 10 of [Neuwirth]). Array equals exp(D), where D is the array with the quadratic sequence (4, 10, 18, 28, ...) on the main subdiagonal and zeros elsewhere.
After adding 1 to the head of the main diagonal and a zero to each of the subdiagonals, the n-th diagonal may be generated as coefficients of (1/n!) [D^(-1) tDt t^(-3)D t^3]^n exp(x*t), where D is the derivative w.r.t. t and D^(-1) t^j/j! = t^(j + 1)/(j + 1)!. E.g., n = 2 generates 20*x*t^3/3! + 90*x^2*t^4/4! + 252*x^3* t^5/5! + ... . For the general unsigned r-Lah number array, replace the threes by (2*r - 1) in the operator. The e.g.f. of the row polynomials is then exp[D^(-1) tDt t^(-(2*r-1))D t^(2*r - 1)] exp(x*t), with offset n = 0. - Tom Copeland, Sep 21 2008

A216294 Triangular array read by rows: T(n,k) is the number of partial permutations of {1,2,...,n} that have exactly k cycles, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 13, 14, 6, 1, 73, 84, 41, 10, 1, 501, 609, 325, 95, 15, 1, 4051, 5155, 2944, 965, 190, 21, 1, 37633, 49790, 30023, 10689, 2415, 343, 28, 1, 394353, 539616, 340402, 129220, 32179, 5348, 574, 36, 1, 4596553, 6478521, 4246842, 1698374, 455511, 84567, 10794, 906, 45, 1

Offset: 0

Geoffrey Critzer, Sep 04 2012

A partial permutation on a set X is a bijection between two subsets of X.
Row sums are A002720.
First column (corresponding to k=0) is A000262.

			1;
1,     1;
3,     3,   1;
13,   14,   6,  1;
73,   84,  41, 10,  1;
501, 609, 325, 95, 15,  1;

Alois P. Heinz, Rows n = 0..50, flattened
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.

Cf. A021009. A000262, A002720, A052852, A062147, A062192, A062266, A088699.

gf := exp(x / (1 - x)) / (1 - x)^y:
serx := series(gf, x, 10): poly := n -> simplify(coeff(serx, x, n)):
seq(print(seq(n!*coeff(poly(n), y, k), k = 0..n)), n = 0..9); # Peter Luschny, Feb 23 2023

nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[Series[Exp[ x/(1-x)]/(1-x)^y,{x,0,nn}],{x,y}]]//Flatten

E.g.f.: exp(x/(1-x))/(1-x)^y.
From Peter Bala, Aug 23 2013: (Start)
Exponential Riordan array [exp(x/(1-x)), log(1/(1-x))].
The row polynomials R(n,y), n > = 0, satisfy the 2nd order recurrence equation R(n,y) = (2*n + y - 1)*R(n-1,y) - (n - 1)*(n + y - 2)*R(n-2,y) with R(0,y) = 1 and R(1,y) = 1 + y.
Modulo variations in offset we have: R(n,0) = A000262, R(n,1) = A002720, R(n,2) = A000262, R(n,3) = A052852, R(n,4) = A062147, R(n,5) = A062266 and R(n,6) = A062192. In general, for fixed k, the sequence {R(n,k)}n>=1 gives the entries on a diagonal of the square array A088699. (End)

A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 13, 1, 4, 13, 34, 73, 1, 5, 21, 73, 209, 501, 1, 6, 31, 136, 501, 1546, 4051, 1, 7, 43, 229, 1045, 4051, 13327, 37633, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 394353, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 4596553

Offset: 0

Seiichi Manyama, Oct 21 2017

			Square array begins:
    1,    1,    1,    1,     1, ... A000012;
    1,    2,    3,    4,     5, ... A000027;
    3,    7,   13,   21,    31, ... A002061;
   13,   34,   73,  136,   229, ... A135859;
   73,  209,  501, 1045,  1961, ...
  501, 1546, 4051, 9276, 19081, ...
Antidiagonal rows begin as:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7, 13;
  1, 4, 13, 34,  73;
  1, 5, 21, 73, 209, 501; - _G. C. Greubel_, Mar 09 2021

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Columns k=0..6 give: A000262, A002720, A000262(n+1), A052852(n+1), A062147, A062266, A062192.
Main diagonal gives A152059.
Similar table: A086885, A088699, A176120.

function t(n,k)
  if n eq 0 then return 1;
  else return Factorial(n-1)*(&+[(j+k)*t(n-j,k)/Factorial(n-j): j in [1..n]]);
  end if; return t;
end function;
[t(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

t[n_, k_]:= t[n, k]= If[n==0, 1, (n-1)!*Sum[(j+k)*t[n-j,k]/(n-j)!, {j,n}]];
T[n_,k_]:= t[k,n-k]; Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

@CachedFunction
def t(n,k): return 1 if n==0 else factorial(n-1)*sum( (j+k)*t(n-j,k)/factorial(n-j) for j in (1..n) )
def T(n,k): return t(k,n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} (j+k)*A(n-j,k)/(n-j)! for n > 0.
A(0,k) = 1, A(1,k) = k+1 and A(n,k) = (2*n-1+k)*A(n-1,k) - (n-1)*(n-2+k)*A(n-2,k) for n > 1.
From Seiichi Manyama, Jan 25 2025: (Start)
A(n,k) = n! * Sum_{j=0..n} binomial(n+k-1,j)/(n-j)!.
A(n,k) = n! * LaguerreL(n, k-1, -1). (End)

A349776 Triangle read by rows: T(n,k) is the number of partitions of set [n] into a set of at most k lists, with 0 <= k <= n. Also called broken permutations.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 6, 12, 13, 0, 24, 60, 72, 73, 0, 120, 360, 480, 500, 501, 0, 720, 2520, 3720, 4020, 4050, 4051, 0, 5040, 20160, 32760, 36960, 37590, 37632, 37633, 0, 40320, 181440, 322560, 381360, 393120, 394296, 394352, 394353

Offset: 0

Ron L.J. van den Burg, Nov 29 2021

List means an ordered subset.

			For n=3 the T(3, 2)=12 broken permutations are {(1, 2, 3)}, {(1, 3, 2)}, {(2, 1, 3)}, {(2, 3, 1)}, {(3, 1, 2)}, {(3, 2, 1)}, {(1, 2), (3)}, {(2, 1), (3)}, {(1, 3), (2)}, {(3, 1), (2)}, {(2, 3), (1)}, {(3, 2), (1)}.
If you add the set of 3 lists {(1), (2), (3)}, you get T(3, 3) = 13 = A000262(3).
Triangle begins:
  1;
  0,   1;
  0,   2,    3;
  0,   6,   12,   13;
  0,  24,   60,   72,   73;
  0, 120,  360,  480,  500,  501;
  0, 720, 2520, 3720, 4020, 4050, 4051;
  ...

Kenneth P. Bogart, Combinatorics Through Guided Discovery, Kenneth P. Bogart, 2004, 57-58.

David Callan, Sets, Lists and Noncrossing Partitions, arXiv:0711.4841 [math.CO], 2007-2008. Also Sets, Lists and Noncrossing Partitions, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.3.
Wikipedia, The twentyfold way

Columns k=0-2 give (for n>=k): A000007, A000142, A001710.
Partial sums of A271703 per row.
Main diagonal is A000262.
Row sums give A062147(n-1) for n>=1.
Cf. A096965.

T:= proc(n, k) option remember; `if`(k<0, 0,
      binomial(n-1, k-1)*n!/k! +T(n, k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 01 2021

T[n_, k_] := Sum[Binomial[n-1, n-j] n!/j!, {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2023 *)

Lah(n, k) = if (n==0, 1, binomial(n-1, k-1)*n!/k!); \\ A271703
T(n, k) = sum(j=0, k, Lah(n, j)); \\ Michel Marcus, Nov 30 2021

def T(n, k):
    return sum(binomial(n, i)*falling_factorial(n-1, n-i) for i in (0..k))
print([[T(n, k) for k in (0..n)] for n in (0..9)])  # Peter Luschny, Dec 01 2021

T(n, k) = Sum_{j=0..k} A271703(n, j) for n >= 0.
T(n, n) = A000262(n).
T(n, k) = Sum_{j=0..k} binomial(n-1, n-j)*n!/j!.
T(n, k) = A000262(n) - A271703(n, k + 1) * hypergeom([1, k - n + 1], [k + 1, k + 2], -1). - Peter Luschny, Nov 30 2021
|Sum_{k=0..n} (-1)^k * T(n,k)| = A096965(n). - Alois P. Heinz, Dec 01 2021

cp's OEIS Frontend

A062137 Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x).

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A086885 Lower triangular matrix, read by rows: T(i,j) = number of ways i seats can be occupied by any number k (0<=k<=j<=i) of persons.

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A143497 Triangle of unsigned 2-Lah numbers.

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A216294 Triangular array read by rows: T(n,k) is the number of partial permutations of {1,2,...,n} that have exactly k cycles, 0<=k<=n.

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A293985 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1-x))/(1-x)^k.

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A349776 Triangle read by rows: T(n,k) is the number of partitions of set [n] into a set of at most k lists, with 0 <= k <= n. Also called broken permutations.

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