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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!. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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)
LINKS
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)
EXAMPLE
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).
MATHEMATICA
Table[k n!, {n, 0, 8}, {k, n + 1}] // Flatten (* Michael De Vlieger, Jul 03 2015 *)
PROG
(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
CROSSREFS
Cf. A178883. - Alford Arnold, Sep 28 2010
Sequence in context: A092991 A355649 A102425 * A143216 A086536 A053045
KEYWORD
easy,nonn,tabl
AUTHOR
Omar E. Pol, Jul 22 2009
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)