A052779 -id:A052779 - cp's OEIS frontend

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

Peter Luschny, May 20 2013

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

			[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

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

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

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

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

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;

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 *)

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

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

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

A129841 Antidiagonal sums of triangle T defined in A048594: T(j,k) = k! * Stirling1(j,k), 1<= k <= j.

Original entry on oeis.org

1, -1, 4, -12, 52, -256, 1502, -10158, 78360, -680280, 6574872, -70075416, 816909816, -10342968456, 141357740736, -2074340369088, 32530886655168, -542971977209760, 9610316495698416, -179788450082431536, 3544714566466060032

Offset: 1

Paul Curtz, May 22 2007

sign

			First seven rows of T are
[    1 ]
[   -1,      2 ]
[    2,     -6,      6 ]
[   -6,     22,    -36,     24 ]
[   24,   -100,    210,   -240,    120 ]
[ -120,    548,  -1350,   2040,  -1800,    720 ]
[  720,  -3528,   9744, -17640,  21000, -15120,   5040 ]

P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, 1969, 135 pages, p. 61. Available from Centre d'Electronique de L'Armement, 35170 Bruz, France, or INRIA, Projets Algorithmes, 78150 Rocquencourt.
P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.
P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, no. 55, pp. 67-78.

Cf. A048594 (T read by rows), A075181 (T unsigned with rows read backwards), A006252 (row sums of T), A000142 (main diagonal of T), A001286 (unsigned first subdiagonal of T). Unsigned values of second through sixth column of T are in A052517, A052748, A052753, A052767, A052779 resp.

m:=21; T:=[ [ Factorial(k)*StirlingFirst(j, k): k in [1..j] ]: j in [1..m] ]; [ &+[ T[j-k+1][k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jun 03 2007

m = 21; t[j_, k_] := k!*StirlingS1[j, k]; Total /@ Table[ t[j-k+1, k], {j, 1, m}, {k, 1, Quotient[j+1, 2]}] (* Jean-François Alcover, Aug 13 2012, translated from Klaus Brockhaus's Magma program *)

E.g.f. for k-th column (k>=1): log(1+x)^k. For further formulas see the references.

Edited and extended by Klaus Brockhaus, Jun 03 2007

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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.

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