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A225479 Triangle read by rows, the ordered Stirling cycle numbers, T(n, k) = k!* s(n, k); n >= 0 k >= 0. 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
The Digital Library of Mathematical Functions defines the Stirling cycle numbers as (-1)^(n-k) times the Stirling numbers of the first kind.
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
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
EXAMPLE
[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
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 *)
PROG
(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)]
(PARI) T(n, k)={k!*abs(stirling(n, k, 1))} \\ Andrew Howroyd, Jul 27 2020
CROSSREFS
Cf. A048594 (signed version without the first column), A132393.
Sequence in context: A355260 A291799 A295027 * A156815 A303439 A303345
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 20 2013
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)