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%I #83 Nov 30 2023 14:44:28
%S 1,0,1,0,2,1,0,6,6,1,0,24,36,12,1,0,120,240,120,20,1,0,720,1800,1200,
%T 300,30,1,0,5040,15120,12600,4200,630,42,1,0,40320,141120,141120,
%U 58800,11760,1176,56,1,0,362880,1451520,1693440,846720,211680,28224,2016,72,1
%N Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.
%C The Lah numbers can be seen as the case m=1 of the family of triangles T_{m}(n,k) = T_{m}(n-1,k-1)+(k^m+(n-1)^m)*T_{m}(n-1,k) (see the link 'Partition transform').
%C This is the Sheffer triangle (lower triangular infinite matrix) (1, x/(1-x)), an element of the Jabotinsky subgroup of the Sheffer group. - _Wolfdieter Lang_, Jun 12 2017
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., pp. 312, 552.
%D I. Lah, Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik, Mitt.-Bl. Math. Statistik, 7:203-213, 1955.
%D T. Mansour, M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, CRC Press, 2016
%H Michael De Vlieger, <a href="/A271703/b271703.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Celeste/celeste3.html"> Two Approaches to Normal Order Coefficients</a>. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
%H M. F. Hasler and P. Luschny, <a href="/wiki/A271703/formulas">Formulas for A271703</a>, OEIS Wiki, Aug. 2017.
%H S. A. Joni, G.-C. Rota, and B. Sagan, <a href="http://dx.doi.org/10.1016/0012-365X(81)90219-3">From sets to functions: Three elementary examples</a>, Discrete Mathematics, Volume 37, Issues 2-3, 1981, 193-202.
%H Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H D. E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, Mathematica J. 2.1 (1992), no. 4, 67-78.
%H Peter Luschny, <a href="http://luschny.de/math/Lah/">Lah numbers</a>
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform#.E2.99.A6.C2.A0_Stirling_and_Lah_numbers_of_higher_order">Partition transform</a>
%H Robert S. Maier, <a href="https://arxiv.org/abs/2308.10332">Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers</a>, arXiv:2308.10332 [math.CO], 2023. See. p. 18.
%H Piotr Miska and Maciej Ulas, <a href="https://arxiv.org/abs/1904.03395">On some properties of the number of permutations being products of pairwise disjoint d-cycles</a>, arXiv:1904.03395 [math.NT], 2019.
%H Emanuele Munarini, <a href="https://doi.org/10.2298/AADM180226017M">Combinatorial identities involving the central coefficients of a Sheffer matrix</a>, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
%F For a collection of formulas see the 'Lah numbers' link.
%F T(n, k) = A097805(n, k)*n!/k! = (-1)^k*P_{n, k}(1,1,1,...) where P_{n, k}(s) is the partition transform of s.
%F T(n, k) = coeff(n! * P(n), x, k) with P(n) = (1/n)*(Sum_{k=0..n-1}(x(n-k)*P(k))), for n >= 1 and P(n=0) = 1, with x(n) = n*x. See A036039. - _Johannes W. Meijer_, Jul 08 2016
%F From _Wolfdieter Lang_, Jun 12 2017: (Start)
%F E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k (that is egf of the triangle) is exp(x*t/(1-t)) (a Sheffer triangle of the Jabotinsky type).
%F E.g.f. column k: (t/(1-t))^k/k!.
%F Three term recurrence: T(n, k) = T(n-1, k-1) + (n-1+k)*T(n, k-1), n >= 1, k = 0..n, with T(0, 0) =1, T(n, -1) = 0, T(n, k) = 0 if n < k.
%F T(n, k) = binomial(n, k)*fallfac(x=n-1, n-k), with fallfac(x, n) = Product_{j=0..(n-1)} (x - j), for n >= 1, and 0 for n = 0.
%F risefac(x, n) = Sum_{k=0..n} T(n, k)*fallfac(k), with risefac(x, n) = Product_{j=0..(n-1)} (x + j), for n >= 1, and 0 for n = 0.
%F See Graham et al., exercise 31, p. 312, solution p. 552. (End)
%F Formally, let f_n(x) = Sum_{k>n} (k-1)!*x^k; then f_n(x) = Sum_{k=0..n} T(n, k)* x^(n+k)*f_0^((k))(x), where ^((k)) stands for the k-th derivative. - _Luc Rousseau_, Dec 27 2020
%F T(n, k) = Sum_{j=k..n} A354795(n, j)*A360177(j, k). - _Mélika Tebni_, Feb 02 2023
%F T(n, k) = binomial(n, k)*(n-1)!/(k-1)! for n, k > 0. - _Chai Wah Wu_, Nov 30 2023
%e As a rectangular array (diagonals of the triangle):
%e 1, 1, 1, 1, 1, 1, ... A000012
%e 0, 2, 6, 12, 20, 30, ... A002378
%e 0, 6, 36, 120, 300, 630, ... A083374
%e 0, 24, 240, 1200, 4200, 11760, ... A253285
%e 0, 120, 1800, 12600, 58800, 211680, ...
%e 0, 720, 15120, 141120, 846720, 3810240, ...
%e A000007, A000142, A001286, A001754, A001755, A001777.
%e The triangle T(n,k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 1
%e 1: 0 1
%e 2: 0 2 1
%e 3: 0 6 6 1
%e 4: 0 24 36 12 1
%e 5: 0 120 240 120 20 1
%e 6: 0 720 1800 1200 300 30 1
%e 7: 0 5040 15120 12600 4200 630 42 1
%e 8: 0 40320 141120 141120 58800 11760 1176 56 1
%e 9: 0 362880 1451520 1693440 846720 211680 28224 2016 72 1
%e 10: 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1
%e ... - _Wolfdieter Lang_, Jun 12 2017
%p T := (n, k) -> `if`(n=k, 1, binomial(n-1,k-1)*n!/k!):
%p seq(seq(T(n, k), k=0..n), n=0..9);
%t T[n_, k_] := Binomial[n, k]*FactorialPower[n-1, n-k];
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 20 2017 *)
%o (Sage)
%o @cached_function
%o def T(n,k):
%o if k<0 : return 0
%o if k==n: return 1
%o return T(n-1,k-1) + (k+n-1)*T(n-1,k)
%o for n in (0..8): print([T(n,k) for k in (0..n)])
%Y Variants: A008297 the main entry for these numbers, A105278, A111596 (signed).
%Y A000262 (row sums). Largest number of the n-th row in A002868.
%Y Cf. A097805, A354795, A360177.
%K nonn,easy,tabl
%O 0,5
%A _Peter Luschny_, Apr 14 2016
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