A051683 Triangle read by rows: T(n,k) = n!*k.

1, 2, 4, 6, 12, 18, 24, 48, 72, 96, 120, 240, 360, 480, 600, 720, 1440, 2160, 2880, 3600, 4320, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 80640, 120960, 161280, 201600, 241920, 282240, 322560, 362880, 725760, 1088640, 1451520, 1814400, 2177280, 2540160, 2903040, 3265920

Offset

1,2

Comments

Numbers with only one nonzero digit when written in factorial base. - Franklin T. Adams-Watters, Nov 28 2011

In other words, numbers m such that A034968(m) = A099563(m). - Antti Karttunen, Jul 02 2013

When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the right within an interval. The subsequence A001563 denotes the circular shifts that start with the first element. Compare A211370 for circular shifts to the left. - Tilman Piesk, Apr 29 2017

Links

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened

Tilman Piesk, Circular shifts to the right (Arrays of permutations).

Formula

T(n,k) = A000142(A002024(n)) * A002260(n,k) = A002024(n)! * A002260(n,k) - Antti Karttunen, Jul 02 2013

Sum_{n>=1} 1/a(n) = e * (gamma - Ei(-1)) = A347952. - Amiram Eldar, Oct 13 2024

Example

Table begins
   1;
   2,  4;
   6, 12, 18;
  24, 48, 72, 96; ...

Mathematica

T[n_, k_] := n!*k; Flatten[Table[T[n, k], {n, 9}, {k, n}]] (* Jean-François Alcover, Apr 22 2011 *)

Prog

(Haskell)
a051683 n k = a051683_tabl !! (n-1) !! (k-1)
a051683_row n = a051683_tabl !! (n-1)
a051683_tabl = map fst $ iterate f ([1], 2) where
   f (row, n) = (row' ++ [head row' + last row'], n + 1) where
     row' = map (* n) row
-- Reinhard Zumkeller, Mar 09 2012
(Scheme): (define (A051683 n) (* (A000142 (A002024 n)) (A002260 n))) -- Antti Karttunen, Jul 02 2013
(Magma) [[Factorial(n)*k: k in [1..n]]: n in [1..15]]; // Vincenzo Librandi, Jun 15 2015
(PARI) for(n=1, 10, for(k=1, n, print1(n!*k, ", "))) \\ G. C. Greubel, Mar 27 2018

Crossrefs

Cf. A000142, A002024, A002260, row sums give A001286(n+1).

Cf. A001563, A007623, A034968, A099563, A200748, A162608, A347952.

Sequence in context: A181416 A225566 A273009 * A215821 A332284 A192096

Adjacent sequences: A051680 A051681 A051682 * A051684 A051685 A051686

Keyword

easy,nice,nonn,tabl

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Status

approved