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.

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, 300, 30, 1, 0, 5040, 15120, 12600, 4200, 630, 42, 1, 0, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, 0, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1

Offset

0,5

Comments

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').

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

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., pp. 312, 552.

I. Lah, Eine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik, Mitt.-Bl. Math. Statistik, 7:203-213, 1955.

T. Mansour, M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, CRC Press, 2016

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

M. F. Hasler and P. Luschny, Formulas for A271703, OEIS Wiki, Aug. 2017.

S. A. Joni, G.-C. Rota, and B. Sagan, From sets to functions: Three elementary examples, Discrete Mathematics, Volume 37, Issues 2-3, 1981, 193-202.

Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.

D. E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78.

Peter Luschny, Lah numbers

Peter Luschny, Partition transform

Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See. p. 18.

Piotr Miska and Maciej Ulas, On some properties of the number of permutations being products of pairwise disjoint d-cycles, arXiv:1904.03395 [math.NT], 2019.

Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Formula

For a collection of formulas see the 'Lah numbers' link.

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.

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

From Wolfdieter Lang, Jun 12 2017: (Start)

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).

E.g.f. column k: (t/(1-t))^k/k!.

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.

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.

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.

See Graham et al., exercise 31, p. 312, solution p. 552. (End)

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

T(n, k) = Sum_{j=k..n} A354795(n, j)*A360177(j, k). - Mélika Tebni, Feb 02 2023

T(n, k) = binomial(n, k)*(n-1)!/(k-1)! for n, k > 0. - Chai Wah Wu, Nov 30 2023

Example

As a rectangular array (diagonals of the triangle):
  1,      1,       1,       1,       1,       1,       ... A000012
  0,      2,       6,       12,      20,      30,      ... A002378
  0,      6,       36,      120,     300,     630,     ... A083374
  0,      24,      240,     1200,    4200,    11760,   ... A253285
  0,      120,     1800,    12600,   58800,   211680,  ...
  0,      720,     15120,   141120,  846720,  3810240, ...
A000007, A000142, A001286, A001754, A001755,  A001777.
The triangle T(n,k) begins:
n\k 0       1        2        3        4       5      6     7    8  9 10 ...
0:  1
1:  0       1
2:  0       2        1
3:  0       6        6        1
4:  0      24       36       12        1
5:  0     120      240      120       20       1
6:  0     720     1800     1200      300      30      1
7:  0    5040    15120    12600     4200     630     42     1
8:  0   40320   141120   141120    58800   11760   1176    56    1
9:  0  362880  1451520  1693440   846720  211680  28224  2016   72  1
10: 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90  1
...  - Wolfdieter Lang, Jun 12 2017

Maple

T := (n, k) -> `if`(n=k, 1, binomial(n-1, k-1)*n!/k!):
seq(seq(T(n, k), k=0..n), n=0..9);

Mathematica

T[n_, k_] := Binomial[n, k]*FactorialPower[n-1, n-k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

Prog

(Sage)
@cached_function
def T(n, k):
    if k<0 : return 0
    if k==n: return 1
    return T(n-1, k-1) + (k+n-1)*T(n-1, k)
for n in (0..8): print([T(n, k) for k in (0..n)])

Crossrefs

Variants: A008297 the main entry for these numbers, A105278, A111596 (signed).

A000262 (row sums). Largest number of the n-th row in A002868.

Cf. A097805, A354795, A360177.

Sequence in context: A352369 A111184 A111596 * A276922 A129062 A281662

Adjacent sequences: A271700 A271701 A271702 * A271704 A271705 A271706

Keyword

nonn,easy,tabl

Author

Peter Luschny, Apr 14 2016

Status

approved