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A271703 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.

%I A271703 #93 Jul 30 2025 11:17:41
%S A271703 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 A271703 300,30,1,0,5040,15120,12600,4200,630,42,1,0,40320,141120,141120,
%U A271703 58800,11760,1176,56,1,0,362880,1451520,1693440,846720,211680,28224,2016,72,1
%N A271703 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 A271703 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 A271703 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 A271703 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., pp. 312, 552.
%D A271703 I. Lah, Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik, Mitt.-Bl. Math. Statistik, 7:203-213, 1955.
%D A271703 T. Mansour, M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, CRC Press, 2016
%H A271703 Michael De Vlieger, <a href="/A271703/b271703.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H A271703 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 A271703 M. F. Hasler and Peter Luschny, <a href="/wiki/A271703/formulas">Formulas for A271703</a>, OEIS Wiki, Aug. 2017.
%H A271703 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 A271703 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 A271703 D. E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, Mathematica J. 2.1 (1992), no. 4, 67-78.
%H A271703 Peter Luschny, <a href="http://luschny.de/math/Lah/">Lah numbers</a>
%H A271703 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 A271703 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 A271703 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 A271703 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.
%H A271703 Elena L. Wang and Guoce Xin, <a href="https://arxiv.org/abs/2507.15654">On Ward Numbers and Increasing Schröder Trees</a>, arXiv:2507.15654 [math.CO], 2025. See p. 12.
%F A271703 For a collection of formulas see the 'Lah numbers' link.
%F A271703 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 A271703 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 A271703 From _Wolfdieter Lang_, Jun 12 2017: (Start)
%F A271703 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 A271703 E.g.f. column k: (t/(1-t))^k/k!.
%F A271703 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 A271703 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 A271703 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 A271703 See Graham et al., exercise 31, p. 312, solution p. 552. (End)
%F A271703 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 A271703 T(n, k) = Sum_{j=k..n} A354795(n, j)*A360177(j, k). - _Mélika Tebni_, Feb 02 2023
%F A271703 T(n, k) = binomial(n, k)*(n-1)!/(k-1)! for n, k > 0. - _Chai Wah Wu_, Nov 30 2023
%e A271703 As a rectangular array (diagonals of the triangle):
%e A271703   1,      1,       1,       1,       1,       1,       ... A000012
%e A271703   0,      2,       6,       12,      20,      30,      ... A002378
%e A271703   0,      6,       36,      120,     300,     630,     ... A083374
%e A271703   0,      24,      240,     1200,    4200,    11760,   ... A253285
%e A271703   0,      120,     1800,    12600,   58800,   211680,  ...
%e A271703   0,      720,     15120,   141120,  846720,  3810240, ...
%e A271703 A000007, A000142, A001286, A001754, A001755,  A001777.
%e A271703 The triangle T(n,k) begins:
%e A271703 n\k 0       1        2        3        4       5      6     7    8  9 10 ...
%e A271703 0:  1
%e A271703 1:  0       1
%e A271703 2:  0       2        1
%e A271703 3:  0       6        6        1
%e A271703 4:  0      24       36       12        1
%e A271703 5:  0     120      240      120       20       1
%e A271703 6:  0     720     1800     1200      300      30      1
%e A271703 7:  0    5040    15120    12600     4200     630     42     1
%e A271703 8:  0   40320   141120   141120    58800   11760   1176    56    1
%e A271703 9:  0  362880  1451520  1693440   846720  211680  28224  2016   72  1
%e A271703 10: 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90  1
%e A271703 ...  - _Wolfdieter Lang_, Jun 12 2017
%p A271703 T := (n, k) -> `if`(n=k, 1, binomial(n-1,k-1)*n!/k!):
%p A271703 seq(seq(T(n, k), k=0..n), n=0..9);
%t A271703 T[n_, k_] := Binomial[n, k]*FactorialPower[n-1, n-k];
%t A271703 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 20 2017 *)
%o A271703 (Sage)
%o A271703 @cached_function
%o A271703 def T(n,k):
%o A271703     if k<0 : return 0
%o A271703     if k==n: return 1
%o A271703     return T(n-1,k-1) + (k+n-1)*T(n-1,k)
%o A271703 for n in (0..8): print([T(n,k) for k in (0..n)])
%Y A271703 Variants: A008297 the main entry for these numbers, A105278, A111596 (signed).
%Y A271703 A000262 (row sums). Largest number of the n-th row in A002868.
%Y A271703 Cf. A097805, A354795, A360177.
%K A271703 nonn,easy,tabl
%O A271703 0,5
%A A271703 _Peter Luschny_, Apr 14 2016