A162608 Triangle read by rows in which row n lists n+1 terms, starting with n!, such that the difference between successive terms is also equal to n!.

1, 1, 2, 2, 4, 6, 6, 12, 18, 24, 24, 48, 72, 96, 120, 120, 240, 360, 480, 600, 720, 720, 1440, 2160, 2880, 3600, 4320, 5040, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 40320, 80640, 120960, 161280, 201600, 241920, 282240, 322560, 362880

Offset: 0

Author

Omar E. Pol, Jul 22 2009

Keywords

• easy
• nonn
• tabl

Comments

Note that the last term of the n-th row is the factorial of (n+1) = (n+1)! = A000142(n+1).

Sequence A178883 (with shape A000041) is a "refinement" of Table A162608; as expected, both sequences have row sums A001710(n+2). - Alford Arnold, Sep 28 2010

From Dennis P. Walsh, May 18 2020: (Start)

T(n,k) provides the number of length (n+2) permutations with elements 1 and 2 as cycle-mates in a (k+1)-cycle. We note that 1 and 2 are cycle-mates if they are elements of the same cycle in the permutation.

For example, T(3,2) counts the 12 permutations of length 5 that have 1 and 2 in the same 3 cycle, namely, (1 2 3)(4)(5), (1 3 2)(4)(5), (1 2 3)(4 5), (1 3 2)(4 5), (1 2 4)(3)(5), (1 4 2)(3)(5), (1 2 4)(3 5), (1 4 2)(3 5),(1 2 5)(3)(4), (1 5 2)(3)(4), (1 2 5)(3 4), and (1 5 2)(3 4).

Note that there are binomial(n,k-1) ways to choose the other (k-1) cycle-mates of 1 and 2 in the (k+1)-cycle and then k! different (k+1)-cycles with these elements. Since there are (n+1-k)! ways to permute the remaining elements, we obtain T(n,k) = (n+1-k)!*k!*binomial(n,k-1) = n!*k. (End)

Examples

			Triangle begins:
1;
1,     2;
2,     4,     6;
6,     12,    18,     24;
24,    48,    72,     96,     120;
120,   240,   360,    480,    600,    720;
720,   1440,  2160,   2880,   3600,   4320,   5040;
5040,  10080, 15120,  20160,  25200,  30240,  35280,  40320;
40320, 80640, 120960, 161280, 201600, 241920, 282240, 322560, 362880;
362880,725760,1088640,1451520,1814400,2177280,2540160,2903040,3265920,3628800;
...
Observation: It appears that rows sums = A001710(n+2).
		

Links

• Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Crossrefs

Cf. A000142, A001710, A051683, A159797, A162611, A162614, A162622.

Cf. A178883. - Alford Arnold, Sep 28 2010

Programs

Haskell

a162608 n k = a162608_tabl !! n !! k
a162608_row n = a162608_tabl !! n
a162608_tabl = map fst $ iterate f ([1], 1) where
   f (row, n) = (row' ++ [head row' + last row'], n + 1) where
     row' = map (* n) row
-- Reinhard Zumkeller, Mar 09 2012

Magma

/* As triangle */ [[Factorial(n)*k: k in [1..n+1]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 04 2015

Mathematica

Table[k n!, {n, 0, 8}, {k, n + 1}] // Flatten (* Michael De Vlieger, Jul 03 2015 *)

Formula

From Robert Israel, Jul 03 2015: (Start)

T(n,k) = n!*k, k = 1 .. n+1.

T(n+1,k) = (n+1)*T(n,k).

T(n,k+1) = T(n,k)+T(n,1). (End)