A143499 Triangle of unsigned 4-Lah numbers.

1, 8, 1, 72, 18, 1, 720, 270, 30, 1, 7920, 3960, 660, 44, 1, 95040, 59400, 13200, 1320, 60, 1, 1235520, 926640, 257400, 34320, 2340, 78, 1, 17297280, 15135120, 5045040, 840840, 76440, 3822, 98, 1, 259459200, 259459200, 100900800, 20180160, 2293200

Offset

4,2

Comments

This is the case r = 4 of the unsigned r-Lah numbers L(r;n,k). The unsigned 4-Lah numbers count the partitions of the set {1,2,...,n} into k ordered lists with the restriction that the elements 1, 2, 3 and 4 belong to different lists. For other cases see A105278 (r = 1), A143497 (r = 2 and comments on the general case) and A143498 (r = 3).

The unsigned 4-Lah numbers are related to the 4-Stirling numbers: the lower triangular array of unsigned 4-Lah numbers may be expressed as the matrix product St1(4) * St2(4), where St1(4) = A143493 and St2(4) = A143496 are the arrays of 4-restricted Stirling numbers of the first and second kind respectively. An alternative factorization for the array is as St1 * P^6 * 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.

Links

Table of n, a(n) for n=4..44.

Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).

G. Nyul, G. Rácz, The r-Lah numbers, Discrete Mathematics, 338 (2015), 1660-1666.

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

Formula

T(n,k) = (n-4)!/(k-4)!*binomial(n+3,k+3), n,k >= 4.

Recurrence: T(n,k) = (n+k-1)*T(n-1,k) + T(n-1,k-1) for n,k >= 4, with the boundary conditions: T(n,k) = 0 if n < 4 or k < 4; T(4,4) = 1.

E.g.f. for column k: Sum_{n >= k} T(n,k)*t^n/(n-4)! = 1/(k-4)!*t^k/(1-t)^(k+4) for k >= 4.

E.g.f: Sum_{n = 4..inf} Sum_{k = 4..n} T(n,k)*x^k*t^n/(n-4)! = (x*t)^4/(1-t)^8*exp(x*t/(1-t)) = (x*t)^4*(1 + (8+x)*t +(72+18*x+x^2)*t^2/2! + ...).

Generalized Lah identity: (x+7)*(x+8)*...*(x+n+2) = Sum_{k = 4..n} T(n,k)*(x-1)*(x-2)*...*(x-k+4).

The polynomials 1/n!*Sum_{k = 4..n+4} T(n+4,k)*(-x)^(k-4) for n >= 0 are the generalized Laguerre polynomials Laguerre(n,7,x).

Array = A132493* A143496 = abs(A008275) * ( A007318 )^6 * A008277 (apply Theorem 10 of [Neuwirth]). Array equals exp(D), where D is the array with the quadratic sequence (8,18,30,44, ... ) on the main subdiagonal and zeros everywhere else.

Example

Triangle begins
n\k|......4......5......6......7......8......9
==============================================
4..|......1
5..|......8......1
6..|.....72.....18......1
7..|....720....270.....30......1
8..|...7920...3960....660.....44......1
9..|..95040..59400..13200...1320.....60......1
...
T(5,4) = 8. The partitions of {1,2,3,4,5} into 4 ordered lists, such that the elements 1, 2, 3 and 4 lie in different lists, are: {1}{2}{3}{4,5} and {1}{2}{3}{5,4}, {1}{2}{4}{3,5} and {1}{2}{4}{5,3}, {1}{3}{4}{2,5} and {1}{3}{4}{5,2}, {2}{3}{4}{1,5} and {2}{3}{4}{5,1}.

Maple

with combinat: T := (n, k) -> (n-4)!/(k-4)!*binomial(n+3, k+3): for n from 4 to 13 do seq(T(n, k), k = 4..n) end do;

Mathematica

T[n_, k_] := (n-4)!/(k-4)!*Binomial[n+3, k+3]; Table[T[n, k], {n, 4, 10}, {k, 4, n}] // Flatten (* Amiram Eldar, Nov 26 2018 *)

Crossrefs

Cf. A007318, A008275, A008277, A049388 (column 4), A105278 (unsigned Lah numbers), A143493, A143496, A143497, A143498.

Sequence in context: A075503 A260040 A051379 * A114152 A347111 A254933

Adjacent sequences: A143496 A143497 A143498 * A143500 A143501 A143502

Keyword

easy,nonn,tabl

Author

Peter Bala, Aug 25 2008

Status

approved