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%I #30 Nov 11 2022 18:35:33
%S 1,6,1,42,14,1,336,168,24,1,3024,2016,432,36,1,30240,25200,7200,900,
%T 50,1,332640,332640,118800,19800,1650,66,1,3991680,4656960,1995840,
%U 415800,46200,2772,84,1,51891840,69189120,34594560,8648640,1201200,96096,4368
%N Triangle of unsigned 3-Lah numbers.
%C For a signed version of this triangle see A062138. This is the case r = 3 of the unsigned r-Lah numbers L(r;n,k). The unsigned 3-Lah numbers count the partitions of the set {1,2,...,n} into k ordered lists with the restriction that the elements 1, 2 and 3 belong to different lists. For other cases see A105278 (r = 1), A143497 (r = 2 and comments on the general case) and A143499 (r = 4).
%C The unsigned 3-Lah numbers are related to the 3-Stirling numbers: the lower triangular array of unsigned 3-Lah numbers may be expressed as the matrix product St1(3) * St2(3), where St1(3) = A143492 and St2(3) = A143495 are the arrays of 3-Stirling numbers of the first and second kind respectively. An alternative factorization for the array is as St1 * P^4 * 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.
%H Erich Neuwirth, <a href="https://doi.org/10.1016/S0012-365X(00)00373-3">Recursively defined combinatorial functions: Extending Galton's board</a>, Discrete Math. 239 No. 1-3, 33-51 (2001).
%H G. Nyul, G. Rácz, <a href="https://doi.org/10.1016/j.disc.2014.03.029">The r-Lah numbers</a>, Discrete Mathematics, 338 (2015), 1660-1666.
%H Michael J. Schlosser and Meesue Yoo, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p31">Elliptic Rook and File Numbers</a>, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.
%H M. Shattuck, <a href="https://arxiv.org/abs/1412.8721">Generalized r-Lah numbers</a>, arXiv:1412.8721 [math.CO], 2014.
%F T(n,k) = (n-3)!/(k-3)!*binomial(n+2,k+2) for n,k >= 3.
%F Recurrence: T(n,k) = (n+k-1)*T(n-1,k) + T(n-1,k-1) for n,k >= 3, with the boundary conditions: T(n,k) = 0 if n < 3 or k < 3; T(3,3) = 1.
%F E.g.f. for column k: Sum_{n >= k} T(n,k)*t^n/(n-3)! = 1/(k-3)!*t^k/(1-t)^(k+3) for k >= 3.
%F E.g.f: Sum_{n = 3..inf} Sum_{k = 3..n} T(n,k)*x^k*t^n/(n-3)! = (x*t)^3/(1-t)^6*exp(x*t/(1-t)) = (x*t)^3*(1 + (6+x)*t +(42+14*x+x^2)*t^2/2! + ... ).
%F Generalized Lah identity: (x+5)*(x+6)*...*(x+n+1) = Sum_{k = 3..n} T(n,k)*(x-1)*(x-2)*...*(x-k+3).
%F The polynomials 1/n!*Sum_{k = 3..n+3} T(n+3,k)*(-x)^(k-3) for n >= 0 are the generalized Laguerre polynomials Laguerre(n,5,x). See A062138.
%F Array = A143492 * A143495 = abs(A008275) * ( A007318 )^4 * A008277 (apply Theorem 10 of [Neuwirth]). Array equals exp(D), where D is the array with the quadratic sequence (6,14,24,36, ... ) on the main subdiagonal and zeros everywhere else.
%e Triangle begins
%e n\k|......3......4......5......6......7......8
%e ==============================================
%e 3..|......1
%e 4..|......6......1
%e 5..|.....42.....14......1
%e 6..|....336....168.....24......1
%e 7..|...3024...2016....432.....36......1
%e 8..|..30240..25200...7200....900.....50......1
%e ...
%e T(4,3) = 6. The partitions of {1,2,3,4} into 3 ordered lists, such that the elements 1, 2 and 3 lie in different lists, are: {1}{2}{3,4} and {1}{2}{4,3}, {1}{3}{2,4} and {1}{3}{4,2}, {2}{3}{1,4} and {2}{3}{4,1}.
%p with combinat: T := (n, k) -> (n-3)!/(k-3)!*binomial(n+2,k+2): for n from 3 to 12 do seq(T(n, k), k = 3..n) end do;
%t T[n_, k_] := (n-3)!/(k-3)!*Binomial[n+2, k+2]; Table[T[n, k], {n, 3, 10}, {k, 3, n}] // Flatten (* _Amiram Eldar_, Nov 26 2018 *)
%o (Magma) /* As triangle */ [[Factorial(n-3)/Factorial(k-3)*Binomial(n+2, k+2): k in [3..n]]: n in [3.. 15]]; // _Vincenzo Librandi_, Nov 27 2018
%Y Cf. A001725 (column 3), A007318, A008275, A008277, A062138, A062148 - A062152 (column 4 to column 8), A062191 (alt. row sums), A062192 (row sums), A105278 (unsigned Lah numbers), A143492, A143495, A143497, A143499.
%K easy,nonn,tabl
%O 3,2
%A _Peter Bala_, Aug 25 2008
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