A125553 Triangle read by rows: T(n,k) = S1(n,k)*2^k, where S1(n,k) is an unsigned Stirling number of the first kind (cf. A008275) (n >= 1, 1 <= k <= n).

2, 2, 4, 4, 12, 8, 12, 44, 48, 16, 48, 200, 280, 160, 32, 240, 1096, 1800, 1360, 480, 64, 1440, 7056, 12992, 11760, 5600, 1344, 128, 10080, 52272, 105056, 108304, 62720, 20608, 3584, 256, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512

Offset

1,1

Comments

Row sums are factorial numbers.

T(n,k) is the number of cycle-colored n-permutations possessing exactly k cycles; two colors are available. - Steven Finch, Nov 19 2021

Subtriangle (for 1<=k<=n) of triangle given by [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [2,0,2,0,2,0,2,0,2,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 05 2007

Also the Bell transform of the sequence "a(n) = 2*(n+1)!/(n+1)". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

Links

Table of n, a(n) for n=1..45.

Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Formula

E.g.f.: 1/(1-x)^(2y). - Geoffrey Critzer, Dec 14 2011

Example

Triangle begins:
  2
  2 4
  4 12 8
  12 44 48 16
  48 200 280 160 32
Triangle [0,1,1,2,2,3,3,...] DELTA [2,0,2,0,2,0,2,...], for 0<=k<=n, begins:
  1;
  0, 2;
  0, 2, 4;
  0, 4, 12, 8;
  0, 12, 44, 48, 16;
  0, 48, 200, 280, 160, 32;

Maple

with(combinat): T:=(n, k)->2^k*abs(stirling1(n, k)): for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Jan 05 2007
A008275 := proc(n, k) if n = 0 and k = 0 then 1 ; elif n = 0 or k = 0 then 0 ; else A008275(n-1, k-1)-(n-1)*A008275(n-1, k) ; fi ; end ; A125553 := proc(n, k) abs(A008275(n, k)*2^k) ; end ; nmax := 10 ; for n from 1 to nmax do for k from 1 to n do printf("%d, ", A125553(n, k)) ; od ; od ; # R. J. Mathar, Jan 12 2007
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 2*(n+1)!/(n+1), 9); # Peter Luschny, Jan 27 2016

Mathematica

Flatten[Table[Table[2^k Abs[StirlingS1[n, k]], {k, 1, n}], {n, 1, 8}]] (* Geoffrey Critzer, Dec 14 2011 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, 2 (n + 1)!/(n + 1)], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Crossrefs

Cf. A008275, A208529.

Sequence in context: A321468 A288044 A081164 * A218147 A226978 A243331

Adjacent sequences: A125550 A125551 A125552 * A125554 A125555 A125556

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Jan 04 2007

Extensions

More terms from R. J. Mathar, Jan 12 2007

Status

approved