A225479 Triangle read by rows, the ordered Stirling cycle numbers, T(n, k) = k!* s(n, k); n >= 0 k >= 0.

1, 0, 1, 0, 1, 2, 0, 2, 6, 6, 0, 6, 22, 36, 24, 0, 24, 100, 210, 240, 120, 0, 120, 548, 1350, 2040, 1800, 720, 0, 720, 3528, 9744, 17640, 21000, 15120, 5040, 0, 5040, 26136, 78792, 162456, 235200, 231840, 141120, 40320, 0, 40320, 219168, 708744, 1614816

Offset: 0

Author

Peter Luschny, May 20 2013

Keywords

• nonn
• tabl

Comments

The Digital Library of Mathematical Functions defines the Stirling cycle numbers as (-1)^(n-k) times the Stirling numbers of the first kind.

Examples

			[n\k][0,   1,   2,    3,    4,    5,   6]
[0]   1,
[1]   0,   1,
[2]   0,   1,   2,
[3]   0,   2,   6,    6,
[4]   0,   6,  22,   36,   24,
[5]   0,  24, 100,  210,  240,  120,
[6]   0, 120, 548, 1350, 2040, 1800, 720.
...
T(4,2) = 22: The table below shows the compositions of 4 into two parts.
n = 4    Composition       Weight     4!*Weight
            3 + 1            1/3         8
            1 + 3            1/3         8
            2 + 2          1/2*1/2       6
                                        = =
                                  total 22
		

References

• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, table 245.

Links

• Vincenzo Librandi, Rows n = 0..50, flattened
• Digital Library of Mathematical Functions, Set Partitions: Stirling Numbers
• S. Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients J. Integer. Seq., Vol. 16 (2013), Article 13.1.3

Crossrefs

Cf. A048594 (signed version without the first column), A132393.

Columns k=0..6 are A000007, A104150, A052517, A052748, A052753, A052767, A052779.

Programs

Maple

A225479 := proc(n, k) option remember;
if k > n or  k < 0 then return(0) fi;
if n = 0 and k = 0 then return(1) fi;
k*A225479(n-1, k-1) + (n-1)*A225479(n-1, k) end;
for n from 0 to 9 do seq(A225479(n, k), k = 0..n) od;

Mathematica

t[n_, k_] := k!*StirlingS1[n, k] // Abs; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 02 2013 *)

PARI

T(n,k)={k!*abs(stirling(n,k,1))} \\ Andrew Howroyd, Jul 27 2020

Sage

def A225479(n, k): return factorial(k)*stirling_number1(n, k)
for n in (0..6): [A225479(n,k) for k in (0..n)]

Formula

For a recursion see the Maple program.

T(n, 0) = A000007; T(n, 1) = A000142; T(n, 2) = A052517.

T(n, 3) = A052748; T(n, n) = A000142; T(n, n-1) = A001286.

row sums = A007840; alternating row sums = A006252.

From Peter Bala, Sep 20 2013: (Start)

E.g.f.: 1/(1 + x*log(1 - t)) = 1 + x*t + (x + 2*x^2)*t^2/2! + (2*x + 6*x^2 + 6*x^3)*t^3/3! + ....

T(n,k) = n!*( the sum of the total weight of the compositions of n into k parts where each part i has weight 1/i ) (see Eger, Theorem 1). An example is given below. (End)

T(n,k) = A132393(n,k) * A000142(k). - Philippe Deléham, Jun 24 2015