A061206 -id:A061206 - 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.

A132159 Lower triangular matrix T(n,j) for double application of an iterated mixed order Laguerre transform inverse to A132014. Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-2-n,x).

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 24, 18, 6, 1, 120, 96, 36, 8, 1, 720, 600, 240, 60, 10, 1, 5040, 4320, 1800, 480, 90, 12, 1, 40320, 35280, 15120, 4200, 840, 126, 14, 1, 362880, 322560, 141120, 40320, 8400, 1344, 168, 16, 1, 3628800, 3265920, 1451520, 423360, 90720, 15120

Offset: 0

Tom Copeland, Nov 01 2007

The matrix operation b = T*a can be characterized several ways in terms of the coefficients a(n) and b(n), their o.g.f.'s A(x) and B(x), or their e.g.f.'s EA(x) and EB(x).
1) b(n) = n! Lag[n,(.)!*Lag[.,a1(.),-1],0], umbrally,
where a1(n) = n! Lag[n,(.)!*Lag[.,a(.),-1],0]
2) b(n) = (-1)^n * n! * Lag(n,a(.),-2-n)
3) b(n) = Sum_{j=0..n} (-1)^j * binomial(n,j) * binomial(-2,j) * j! * a(n-j)
4) b(n) = Sum_{j=0..n} binomial(n,j) * (j+1)! * a(n-j)
5) B(x) = (1-xDx))^(-2) A(x), formally
6) B(x) = Sum_{j>=0} (-1)^j * binomial(-2,j) * (xDx)^j A(x)
= Sum_{j>=0} (j+1) * (xDx)^j A(x)
7) B(x) = Sum_{j>=0} (j+1) * x^j * D^j * x^j A(x)
8) B(x) = Sum_{j>=0} (j+1)! * x^j * Lag(j,-:xD:,0) A(x)
9) EB(x) = Sum_{j>=0} x^j * Lag[j,(.)! * Lag[.,a1(.),-1],0]
10) EB(x) = Sum_{j>=0} Lag[j,a1(.),-1] * (-x)^j / (1-x)^(j+1)
11) EB(x) = Sum_{j>=0} x^n * Sum_{j=0..n} (j+1)!/j! * a(n-j) / (n-j)!
12) EB(x) = Sum_{j>=0} (-x)^j * Lag[j,a(.),-2-j]
13) EB(x) = exp(a(.)*x) / (1-x)^2 = (1-x)^(-2) * EA(x)
14) T = A094587^2 = A132013^(-2) = A132014^(-1)
where Lag(n,x,m) are the Laguerre polynomials of order m, D the derivative w.r.t. x and (:xD:)^j = x^j * D^j. Truncating the D operator series at the j = n term gives an o.g.f. for b(0) through b(n).
c = (1!,2!,3!,4!,...) is the sequence associated to T under the list partition transform and associated operations described in A133314. Thus T(n,k) = binomial(n,k)*c(n-k) . c are also the coefficients in formulas 4 and 8.
The reciprocal sequence to c is d = (1,-2,2,0,0,0,...), so the inverse of T is TI(n,k) = binomial(n,k)*d(n-k) = A132014. (A121757 is the reverse of T.)
These formulas are easily generalized for m applications of the basic operator n! Lag[n,(.)!*Lag[.,a(.),-1],0] by replacing 2 by m in formulas 2, 3, 5, 6, 12, 13 and 14, or (j+1)! by (m-1+j)!/(m-1)! in 4, 8 and 11. For further discussion of repeated applications of T, see A132014.
The row sums of T = [formula 4 with a(n) all 1] = [binomial transform of c] = [coefficients of B(x) with A(x) = 1/(1-x)] = A001339. Therefore the e.g.f. of A001339 = [formula 13 with a(n) all 1] = exp(x)*(1-x)^(-2) = exp(x)*exp[c(.)*x)] = exp[(1+c(.))*x].
Note the reciprocal is 1/{exp[(1+c(.))*x]} = exp(-x)*(1-x)^2 = e.g.f. of signed A002061 with leading 1 removed], which makes A001339 and the signed, shifted A002061 reciprocal arrays under the list partition transform of A133314.
The e.g.f. for the row polynomials (see A132382) implies they form an Appell sequence (see Wikipedia). - Tom Copeland, Dec 03 2013
As noted in item 12 above and reiterated in the Bala formula below, the e.g.f. is e^(x*t)/(1-x)^2, and the Poisson-Charlier polynomials P_n(t,y) have the e.g.f. (1+x)^y e^(-xt) (Feinsilver, p. 5), so the row polynomials R_n(t) of this entry are (-1)^n P_n(t,-2). The associated Appell sequence IR_n(t) that is the umbral compositional inverse of this entry's polynomials has the e.g.f. (1-x)^2 e^(xt), i.e., the e.g.f. of A132014 (noted above), and, therefore, the row polynomials (-1)^n PC(t,2). As umbral compositional inverses, R_n(IR.(t)) = t^n =  IR_n(R.(t)), where, by definition, P.(t)^n = P_n(t), is the umbral evaluation. - Tom Copeland, Jan 15 2016
T(n,k) is the number of ways to place (n-k) rooks in a 2 x (n-1) Ferrers board (or diagram) under the Goldman-Haglund i-row creation rook mode for i=2. Triangular recurrence relation is given by T(n,k) = T(n-1,k-1) + (n+1-k)*T(n-1,k). - Ken Joffaniel M. Gonzales, Jan 21 2016

			First few rows of the triangle are
    1;
    2,  1;
    6,  4,  1;
   24, 18,  6, 1;
  120, 96, 36, 8, 1;

Nathaniel Johnston, Rows 0..100, flattened
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
P. Feinsilver, Lie algebras, representations, and analytic semigroups through dual vector fields
Jay Goldman and James Haglund, Generalized rook polynomials, J. Combin. Theory A 91 (2000), 509-530.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Wikipedia, Appell sequence
Wikipedia, Sheffer sequence

Cf. A008277, A094587, A132013, A132382.

Columns: A000142 (k=0), A001563 (k=1), A001286 (k=2), A005990 (k=3), A061206 (k=4), A062199 (k=5), A062148 (k=6).

a132159 n k = a132159_tabl !! n !! k
a132159_row n = a132159_tabl !! n
a132159_tabl = map reverse a121757_tabl
-- Reinhard Zumkeller, Mar 06 2014

/* As triangle */ [[Binomial(n,k)*Factorial(n-k+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 10 2016

T := proc(n,k) return binomial(n,k)*factorial(n-k+1): end: seq(seq(T(n,k),k=0..n),n=0..10); # Nathaniel Johnston, Sep 28 2011

nn=10;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[Series[Exp[y x]/(1-x)^2,{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Feb 15 2013 *)

flatten([[binomial(n,k)*factorial(n-k+1) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, May 19 2021

T(n,k) = binomial(n,k)*c(n-k).
From Peter Bala, Jul 10 2008: (Start)
T(n,k) = binomial(n,k)*(n-k+1)!.
T(n,k) = (n-k+1)*T(n-1,k) + T(n-1,k-1).
E.g.f.: exp(x*y)/(1-y)^2 = 1 + (2+x)*y + (6+4*x+x^2)*y^2/2! + ... .
This array is the particular case P(2,1) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown below:
  n\k|0....................1...............2.........3.....4
  ----------------------------------------------------------
  0..|1.....................................................
  1..|a....................1................................
  2..|a(a+b)...............2a..............1................
  3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
  4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
  ...
See A094587 for some general properties of these arrays.
Other cases recorded in the database include: P(1,0) = Pascal's triangle A007318, P(1,1) = A094587, P(2,0) = A038207, P(3,0) = A027465, P(1,3) = A136215 and P(2,3) = A136216. (End)
Let f(x) = (1/x^2)*exp(-x). The n-th row polynomial is R(n,x) = (-x)^n/f(x)*(d/dx)^n(f(x)), and satisfies the recurrence equation R(n+1,x) = (x+n+2)*R(n,x)-x*R'(n,x). Cf. A094587. - Peter Bala, Oct 28 2011
Exponential Riordan array [1/(1 - y)^2, y]. The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = Sum_{k=0..n} binomial(n,k)*y^(n-k)*R(k,x). Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = Product_{k = 0..n-1} (2*x + k) with the convention that P(0,x) = 1. Then the present triangle is the triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x). For example, row 3 is (24, 18, 6, 1) so P(3,x + 1) = (2*x + 2)*(2*x + 3)*(2*x + 4) = 24 + 18*(2*x) + 6*(2*x)*(2*x + 1) + (2*x)*(2*x + 1)*(2*x + 2). Matrix square of triangle A094587. - Peter Bala, Aug 29 2013
From Tom Copeland, Apr 21 2014: (Start)
T = (I-A132440)^(-2) = {2*I - exp[(A238385-I)]}^(-2) = unsigned exp[2*(I-A238385)] = exp[A005649(.)*(A238385-I)], umbrally, where I = identity matrix.
The e.g.f. is exp(x*y)*(1-y)^(-2), so the row polynomials form an Appell sequence with lowering operator D=d/dx and raising operator x+2/(1-D).
With L(n,m,x) = Laguerre polynomials of order m, the row polynomials are (-1)^n * n! * L(n,-2-n,x) = (-1)^n*(-2!/(-2-n)!)*K(-n,-2-n+1,x) where K is Kummer's confluent hypergeometric function (as a limit of n+s as s tends to zero).
Operationally, (-1)^n*n!*L(n,-2-n,-:xD:) = (-1)^n*x^(n+2)*:Dx:^n*x^(-2-n) = (-1)^n*x^2*:xD:^n*x^(-2) = (-1)^n*n!*binomial(xD-2,n) = (-1)^n*n!*binomial(-2,n)*K(-n,-2-n+1,-:xD:) where :AB:^n = A^n*B^n for any two operators. Cf. A235706.
The generalized Pascal triangle Bala mentions is a special case of the fundamental generalized factorial matrices in A133314. (End)
From Peter Bala, Jul 26 2021: (Start)
O.g.f: 1/y * Sum_{k >= 0} k!*( y/(1 - x*y) )^k =  1 + (2 + x)*y + (6 + 4*x + x^2)*y^2 + ....
First-order recurrence for the row polynomials: (n - x)*R(n,x) = n*(n - x + 1)*R(n-1,x) - x^(n+1) with R(0,x) = 1.
R(n,x) = (x + n + 1)*R(n-1,x) - (n - 1)*x*R(n-2,x) with R(0,x) = 1 and R(1,x) = 2 + x.
R(n,x) = A087981 (x = -2), A000255 (x = -1), A000142 (x = 0), A001339 (x = 1), A081923 (x = 2) and A081924 (x = 3). (End)

Formula 3) in comments corrected by Tom Copeland, Apr 20 2014

Title modified by Tom Copeland, Apr 23 2014

A264173 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 1324; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 110, 10, 632, 86, 2, 4229, 782, 29, 32337, 7571, 407, 5, 278204, 78726, 5856, 94, 2659223, 882997, 84351, 2215, 14, 27959880, 10657118, 1251246, 48234, 322, 320706444, 137977980, 19318314, 984498, 14322, 42, 3985116699, 1910131680, 311306106

Offset: 0

Alois P. Heinz, Nov 06 2015

Pattern 4231 gives the same triangle.

			T(4,1) = 1: 1324.
T(6,2) = 2: 132546, 142536.
T(8,3) = 5: 13254768, 13264758, 14253768, 14263758, 15263748.
T(10,4) = 14: 132547698(10), 132548697(10), 132647598(10), 132648597(10), 132748596(10), 142537698(10), 142538697(10), 142637598(10), 142638597(10), 142738596(10), 152637498(10), 152638497(10), 152738496(10), 162738495(10).
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        2;
03 :        6;
04 :       23,      1;
05 :      110,     10;
06 :      632,     86,     2;
07 :     4229,    782,    29;
08 :    32337,   7571,   407,    5;
09 :   278204,  78726,  5856,   94;
10 :  2659223, 882997, 84351, 2215, 14;

Alois P. Heinz, Rows n = 0..120, flattened

Columns k=0-10 give: A113228, A264174, A264175, A264176, A264177, A264178, A264179, A264180, A264181, A264182, A264183.
Row sums give A000142.
T(2n+2,n) gives A000108(n) for n>0.
Cf. A004526, A061206, A264319 (pattern 3412).

b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
      add(b(u-j, o+j-1, `if`(t>0 and j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);

b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1, Sum[b[u - j, o + j - 1, If[t > 0 && j < t, -j, 0]], {j, 1, u}] + Sum[b[u + j - 1, o - j, j] * If[t < 0 && -j <= t, x, 1], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0, 0]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Apr 30 2017, translated from Maple *)

Sum_{k>0} k * T(n,k) = ceiling((n-3)*n!/4!) = A061206(n-3) (for n>3).

A264319 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 3412; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 110, 10, 631, 88, 1, 4223, 794, 23, 32301, 7639, 379, 1, 277962, 79164, 5706, 48, 2657797, 885128, 84354, 1520, 1, 27954521, 10657588, 1266150, 38452, 89, 320752991, 137752283, 19621124, 869740, 5461, 1, 3987045780, 1904555934, 316459848

Offset: 0

Alois P. Heinz, Nov 11 2015

Pattern 2143 gives the same triangle.

			T(4,1) = 1: 3412.
T(5,1) = 10: 14523, 24513, 34125, 34512, 35124, 43512, 45123, 45132, 45231, 53412.
T(6,2) = 1: 563412.
T(7,2) = 23: 1674523, 2674513, 3674512, 4673512, 5614723, 5624713, 5634127, 5634712, 5673412, 5714623, 5724613, 5734126, 5734612, 6573412, 6714523, 6724513, 6734125, 6734512, 6735124, 6745123, 6745132, 6745231, 7563412.
T(8,3) = 1: 78563412.
T(9,3) = 48: 189674523, 289674513, 389674512, ..., 896745132, 896745231, 978563412.
Triangle T(n,k) begins:
00 :       1;
01 :       1;
02 :       2;
03 :       6;
04 :      23,      1;
05 :     110,     10;
06 :     631,     88,     1;
07 :    4223,    794,    23;
08 :   32301,   7639,   379,    1;
09 :  277962,  79164,  5706,   48;
10 : 2657797, 885128, 84354, 1520, 1;

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

Columns k=0-10 give: A113229, A264320, A264321, A264322, A264323, A264324, A264325, A264326, A264327, A264328, A264329.
Row sums give A000142.
Cf. A004526, A061206, A264173 (pattern 1324).

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(expand(
       b(u+j-1, o-j, j)*`if`(t<0 and j<1-t, x, 1)), j=1..o)+
      add(b(u-j, o+j-1, `if`(t>0 and j>t, t-j, 0)), j=1..u))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[Expand[b[u+j-1, o-j, j]*If[t<0 && j<1-t, x, 1]], {j, 1, o}] + Sum[b[u-j, o+j-1, If[t>0 && j>t, t-j, 0]], {j, 1, u}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 16 2017, translated from Maple_ *)

Sum_{k>0} k * T(n,k) = ceiling((n-3)*n!/4!) = A061206(n-3) (for n>3).

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

A062141 Third column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

Original entry on oeis.org

1, 18, 252, 3360, 45360, 635040, 9313920, 143700480, 2335132800, 39956716800, 719220902400, 13599813427200, 269729632972800, 5602076992512000, 121645100408832000, 2757288942600192000, 65140951268929536000, 1601701037083090944000, 40932359836567879680000

Offset: 0

Wolfdieter Lang, Jun 19 2001

			a(2) = (2+2)! * binomial(2+5,5) / 2! = (24*21)/2 = 252. - _Indranil Ghosh_, Feb 24 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Index entries for sequences related to Laguerre polynomials

Cf. A061206, A062137.

[Factorial(n+2)*Binomial(n+5,5)/2: n in [0..20]]; // G. C. Greubel, May 11 2018

a:= n->(n+2)!*binomial(n+5, 5)/2!: seq(a(n), n=0..20); # Zerinvary Lajos, Apr 29 2007

Table[(n+2)!*Binomial[n+5,5]/2!,{n,0,15}] (* Indranil Ghosh, Feb 24 2017 *)

a(n)=(n+2)!*binomial(n+5,5)/2! \\ Indranil Ghosh, Feb 24 2017

x='x+O('x^30); Vec(serlaplace((1+10*x+10*x^2)/(1-x)^8)) \\ G. C. Greubel, May 11 2018

import math
f=math.factorial
def C(n,r):return f(n)/f(r)/f(n-r)
def A062141(n): return f(n+2)*C(n+5,5)/f(2) # Indranil Ghosh, Feb 24 2017

[binomial(n,5)*factorial (n-3)/2 for n in range(5, 21)] # Zerinvary Lajos, Jul 07 2009

a(n) = (n+2)!*binomial(n+5, 5)/2!.
E.g.f.: (1 + 10*x + 10*x^2)/(1-x)^8.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j), then a(n-5) = (-1)^(n-1)*f(n,5,-3), (n >= 5). - Milan Janjic, Mar 01 2009

A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 6, 18, 24, 18, 6, 1, 1, 8, 36, 96, 120, 96, 36, 8, 1, 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1, 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 35280, 15120, 4200, 840, 126, 14, 1

Offset: 1

Alois P. Heinz, Feb 18 2019

			Triangle T(n,k) begins:
  :                                 1                              ;
  :                           1,    2,    1                        ;
  :                     1,    4,    6,    4,    1                  ;
  :               1,    6,   18,   24,   18,    6,   1             ;
  :          1,   8,   36,   96,  120,   96,   36,   8,  1         ;
  :      1, 10,  60,  240,  600,  720,  600,  240,  60, 10,  1     ;
  :  1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1  ;

Alois P. Heinz, Rows n = 1..100, flattened
Wikipedia, Permutation
Wikipedia, Permutation matrix

Columns k=0-6 give (offsets may differ): A000142, A001563, A001286, A005990, A061206, A062199, A062148.
Row sums give A306495(n-1).
Cf. A132159 (right part of triangle), A306234, A324225.

b:= proc(s, c) option remember; (n-> `if`(n=0, c,
      add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t

T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)

T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)!/t! if t < n with t = |k|, T(n,k) = 0 otherwise.
T(n,k) = 1/|k|! * A324225(n,k).
E.g.f. of column k: x^t/t! * hypergeom([2, t], [t+1], x) with t = |k|+1.
Sum_{k=1-n..n-1} T(n,k) = A306495(n-1).

A138770 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).

Original entry on oeis.org

2, 4, 2, 12, 8, 4, 48, 36, 24, 12, 240, 192, 144, 96, 48, 1440, 1200, 960, 720, 480, 240, 10080, 8640, 7200, 5760, 4320, 2880, 1440, 80640, 70560, 60480, 50400, 40320, 30240, 20160, 10080, 725760, 645120, 564480, 483840, 403200, 322560, 241920, 161280, 80640

Offset: 2

Emeric Deutsch, Apr 06 2008

Sum of row n = n! = A000142(n).

The expected value of k is (n-2)/3. [Geoffrey Critzer, Dec 19 2009]

			T(4,2)=4 because we have 1342, 1432, 2341 and 2431.
Triangle starts:
  2;
  4,2;
  12,8,4;
  48,36,24,12;
  240,192,144,96,48;
  ...

Cf. A000142, A052489, A052582, A052609, A005990, A061206, A052849, A090672.

T:=proc(n,k) if n-2 < k then 0 else (2*n-2*k-2)*factorial(n-2) end if end proc; for n from 2 to 10 do seq(T(n, k),k=0..n-2) end do; # yields sequence in triangular form

Table[Table[2 (n - r) (n - 2)!, {r, 1, n - 1}], {n, 1, 10}] // Grid (* Geoffrey Critzer, Dec 19 2009 *)

T(n,k) = 2*(n-k-1)*(n-2)!.
T(n,0) = 2(n-1)! = A052849(n-1).
T(n,1) = A052582(n-2).
T(n,2) = A052609(n-2).
T(n,3) = 12*A005990(n-3).
T(n,4) = 48*A061206(n-5).
T(n,n-2) = 2(n-2)! (A052849).
Sum_{k=0..n-2} k*T(n,k) = n!*(n-2)/3 = A090672(n-1).

A216118 Triangle read by rows: T(n,k) is the number of stretching pairs in all permutations in S_{n,k} (=set of permutations in S_n with k cycles) (n >= 3; 1 <= k <= n-2).

Original entry on oeis.org

0, 1, 1, 10, 15, 5, 90, 165, 90, 15, 840, 1750, 1225, 350, 35, 8400, 19180, 15750, 5950, 1050, 70, 90720, 222264, 204624, 92610, 22050, 2646, 126, 1058400, 2744280, 2757720, 1421490, 411600, 67620, 5880, 210, 13305600, 36162720, 38980920, 22203720, 7408170, 1496880, 180180, 11880, 330

Offset: 3

Emeric Deutsch, Feb 26 2013

A stretching pair of a permutation p in S_n is a pair (i,j) (1 <= i < j <= n) satisfying p(i) < i < j < p(j). For example, for the permutation 31254 in S_5 the pair (2,4) is stretching because 1< p(2) < 2 < 4 < p(4) = 5.
Number of entries in row n (n >= 3) is n - 2.
Sum of entries in row n is A216119(n).
T(n,1) = A061206(n-3).

			T(4,1) = 1, T(4,2) = 1 because 22 permutations in S_4 have no stretching pairs, the 1-cycle 3142 has the stretching pair (2,3) and the 2-cycle 2143 has the stretching pair (2,3).
Triangle starts:
    0;
    1,    1;
   10,   15,    5;
   90,  165,   90,  15;
  840, 1750, 1225, 350, 35;
  ...

E. Lundberg and B. Nagle, A permutation statistic arising in dynamics of internal maps. (submitted)

Muniru A Asiru, Rows n=3..100 of triangle, flattened
E. Clark and R. Ehrenborg, Explicit expressions for the extremal excedance statistic, European J. Combinatorics, 31, 2010, 270-279.
J. Cooper, E. Lundberg, and B. Nagle, Generalized pattern frequency in large permutations, Electron. J. Combin. 20, 2013, #P28.

Cf. A216119, A061206.

List([3..12],n->List([1..n-2],k->Binomial(n,4)*Stirling1(n-2,k))); # Muniru A Asiru, Dec 13 2018

[[(-1)^(n-k)*Binomial(n,4)*StirlingFirst(n-2,k): k in [1..n-2]]: n in [3..12]]; // G. C. Greubel, Dec 13 2018

with(combinat): T := proc (n, k) options operator, arrow: binomial(n, 4)*abs(stirling1(n-2, k)) end proc: for n from 3 to 12 do seq(T(n, k), k = 1 .. n-2) end do; # yields sequence in triangular form

T[n_, k_] := Binomial[n, 4] * Abs[StirlingS1[n-2, k]]; Table[T[n, k], {n, 3, 12}, {k, 1, n-2}] // Flatten (* Amiram Eldar, Dec 13 2018 *)

{T(n,k) = (-1)^(n-k)*binomial(n,4)*stirling(n-2,k,1)};
for(n=3, 10, for(k=1,n-2, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 13 2018

[[binomial(n,4)*stirling_number1(n-2,k) for k in (1..n-2)] for n in (3..12)] # G. C. Greubel, Dec 13 2018

T(n,k) = binomial(n,4)*abs(Stirling1(n-2,k)).

T(n,k) = binomial(n,4)*(-1)^(n-k)*Stirling1(n-2,k).

A216121 Irregular triangle read by rows: T(n,k) is the number of permutations in C_n (= the 1-cycles in S_n) having k stretching pairs.

Original entry on oeis.org

1, 1, 2, 5, 1, 16, 6, 2, 63, 31, 20, 5, 1, 294, 168, 150, 70, 30, 6, 2, 1585, 997, 1072, 691, 423, 171, 75, 20, 5, 1, 9692, 6522, 7882, 6176, 4744, 2612, 1598, 656, 300, 100, 30, 6, 2, 66275, 46891, 61356, 54561, 49013, 32689, 24285, 13429, 7812, 3795, 1759, 651, 263, 75, 20, 5, 1

Offset: 1

Emeric Deutsch, Feb 26 2013

A stretching pair of a permutation p in S_n is a pair (i,j) (1 <= i < j <= n) satisfying p(i) < i < j < p(j). For example, for the permutation 31254 in S_5 the pair (2,4) is stretching because p(2) = 1 < 2 < 4 < p(4) = 5.
Sum of entries in row n is (n-1)! = A000142(n-1).
T(n,0) = A136127(n-1).
Sum_{k>=1} k*T(n,k) = n!*(n-3)/24 = A061206(n-3).

			T(4,1) = 1 because 3142 has 1 stretching pair (2,3); the other five 1-cycles in S_4 have no stretching pairs.
Triangle starts:
    1;
    1;
    2;
    5,   1;
   16,   6,   2;
   63,  31,  20,  5,  1;
  294, 168, 150, 70, 30, 6, 2;
  ...

E. Lundberg and B. Nagle, A permutation statistic arising in dynamics of internal maps. (submitted)

E. Clark and R. Ehrenborg, Explicit expressions for the extremal excedance statistic, European J. Combinatorics, 31, 2010, 270-279.
J. Cooper, E. Lundberg, and B. Nagle, Generalized pattern frequency in large permutations, Electron. J. Combin. 20, 2013, #P28.

Cf. A136127, A000142, A061206, A216120.

n := 7: with(combinat): nrcyc := proc (p) local nrfp, pc: nrfp := proc (p) local ct, j: ct := 0: for j to nops(p) do if p[j] = j then ct := ct+1 else  end if end do: ct end proc: pc := convert(p, disjcyc): nops(pc)+nrfp(p) end proc: nrcyc := proc (p) local nrfp, pc: nrfp := proc (p) local ct, j: ct := 0: for j to nops(p) do if p[j] = j then ct := ct+1 else  end if end do: ct end proc: pc := convert(p, disjcyc): nops(pc)+nrfp(p) end proc: sp := proc (p) local ct, i, j: ct := 0: for i from 2 to nops(p)-2 do for j from i+1 to nops(p)-1 do if p[i] < i and i < j and j < p[j] then ct := ct+1 else  end if end do end do: ct end proc: P[n] := permute(n): C[n] := {}: for j to factorial(n) do if nrcyc(P[n][j]) = 1 then C[n] := `union`(C[n], {P[n][j]}) else  end if end do: sort(add(t^sp(C[n][j]), j = 1 .. factorial(n-1)));

The values of T(n,k) have been found by straightforward counting (with Maple). The Maple program (improvable!) yields the generating polynomial of the specified row n. Within the program, sp(p) is the number of stretching pairs of the permutation p.

More terms from Alois P. Heinz, Apr 15 2017

cp's OEIS Frontend

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

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A132159 Lower triangular matrix T(n,j) for double application of an iterated mixed order Laguerre transform inverse to A132014. Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-2-n,x).

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A264173 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 1324; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.

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A264319 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 3412; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.

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A143497 Triangle of unsigned 2-Lah numbers.

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A062141 Third column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

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