cp's OEIS Frontend

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!.

%I A162608 #38 Sep 08 2022 08:45:46
%S A162608 1,1,2,2,4,6,6,12,18,24,24,48,72,96,120,120,240,360,480,600,720,720,
%T A162608 1440,2160,2880,3600,4320,5040,5040,10080,15120,20160,25200,30240,
%U A162608 35280,40320,40320,80640,120960,161280,201600,241920,282240,322560,362880
%N 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!.
%C A162608 Note that the last term of the n-th row is the factorial of (n+1) = (n+1)! = A000142(n+1).
%C A162608 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
%C A162608 From _Dennis P. Walsh_, May 18 2020: (Start)
%C A162608 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.
%C A162608 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).
%C A162608 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)
%H A162608 Reinhard Zumkeller, <a href="/A162608/b162608.txt">Rows n=0..150 of triangle, flattened</a>
%F A162608 From _Robert Israel_, Jul 03 2015: (Start)
%F A162608 T(n,k) = n!*k, k = 1 .. n+1.
%F A162608 T(n+1,k) = (n+1)*T(n,k).
%F A162608 T(n,k+1) = T(n,k)+T(n,1). (End)
%e A162608 Triangle begins:
%e A162608 1;
%e A162608 1,     2;
%e A162608 2,     4,     6;
%e A162608 6,     12,    18,     24;
%e A162608 24,    48,    72,     96,     120;
%e A162608 120,   240,   360,    480,    600,    720;
%e A162608 720,   1440,  2160,   2880,   3600,   4320,   5040;
%e A162608 5040,  10080, 15120,  20160,  25200,  30240,  35280,  40320;
%e A162608 40320, 80640, 120960, 161280, 201600, 241920, 282240, 322560, 362880;
%e A162608 362880,725760,1088640,1451520,1814400,2177280,2540160,2903040,3265920,3628800;
%e A162608 ...
%e A162608 Observation: It appears that rows sums = A001710(n+2).
%t A162608 Table[k n!, {n, 0, 8}, {k, n + 1}] // Flatten (* _Michael De Vlieger_, Jul 03 2015 *)
%o A162608 (Haskell)
%o A162608 a162608 n k = a162608_tabl !! n !! k
%o A162608 a162608_row n = a162608_tabl !! n
%o A162608 a162608_tabl = map fst $ iterate f ([1], 1) where
%o A162608    f (row, n) = (row' ++ [head row' + last row'], n + 1) where
%o A162608      row' = map (* n) row
%o A162608 -- _Reinhard Zumkeller_, Mar 09 2012
%o A162608 (Magma) /* As triangle */ [[Factorial(n)*k: k in [1..n+1]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jul 04 2015
%Y A162608 Cf. A000142, A001710, A051683, A159797, A162611, A162614, A162622.
%Y A162608 Cf. A178883. - _Alford Arnold_, Sep 28 2010
%K A162608 easy,nonn,tabl
%O A162608 0,3
%A A162608 _Omar E. Pol_, Jul 22 2009