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

Author

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

Keywords

• easy
• nice
• nonn
• tabl

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

Examples

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

Links

• Reinhard Zumkeller, Rows n=1..150 of triangle, flattened
• Tilman Piesk, Circular shifts to the right (Arrays of permutations).

Crossrefs

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

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

Programs

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

Magma

[[Factorial(n)*k: k in [1..n]]: n in [1..15]]; // Vincenzo Librandi, Jun 15 2015

Mathematica

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

PARI

for(n=1,10, for(k=1,n, print1(n!*k, ", "))) \\ G. C. Greubel, Mar 27 2018

Python

from math import isqrt, factorial, comb
def A051683(n): return factorial(a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(n-comb(a,2)) # Chai Wah Wu, Jun 25 2025

Scheme

(define (A051683 n) (* (A000142 (A002024 n)) (A002260 n))) ;; Antti Karttunen, Jul 02 2013

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